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Forum of Mathematics, Sigma (2017), Vol. 5, e5, 49 pagesdoi:10.1017/fms.2017.1 1

NONCROSSING SETS AND A GRASSMANNASSOCIAHEDRON

FRANCISCO SANTOS1, CHRISTIAN STUMP2 and VOLKMAR WELKER3

1 Departamento de Matematicas, Estadıstica y Computacion Universidad de Cantabria,Santander, Spain;

email: [email protected] Institut fur Mathematik, Freie Universitat Berlin, Germany;

email: [email protected] Fachbereich Mathematik und Informatik, Philipps-Universitat Marburg, Germany;

email: [email protected]

Received 19 August 2015; accepted 10 January 2017

Abstract

We study a natural generalization of the noncrossing relation between pairs of elements in [n] tok-tuples in [n] that was first considered by Petersen et al. [J. Algebra 324(5) (2010), 951–969].We give an alternative approach to their result that the flag simplicial complex on

([n]k

)induced

by this relation is a regular, unimodular and flag triangulation of the order polytope of the posetgiven by the product [k] × [n − k] of two chains (also called Gelfand–Tsetlin polytope), and that itis the join of a simplex and a sphere (that is, it is a Gorenstein triangulation). We then observe thatthis already implies the existence of a flag simplicial polytope generalizing the dual associahedron,whose Stanley–Reisner ideal is an initial ideal of the Grassmann–Plucker ideal, while previousconstructions of such a polytope did not guarantee flagness nor reduced to the dual associahedronfor k = 2. On our way we provide general results about order polytopes and their triangulations.We call the simplicial complex the noncrossing complex, and the polytope derived from it thedual Grassmann associahedron. We extend results of Petersen et al. [J. Algebra 324(5) (2010),951–969] showing that the noncrossing complex and the Grassmann associahedron naturally reflectthe relations between Grassmannians with different parameters, in particular the isomorphismGk,n

∼= Gn−k,n . Moreover, our approach allows us to show that the adjacency graph of thenoncrossing complex admits a natural acyclic orientation that allows us to define a Grassmann–Tamari order on maximal noncrossing families. Finally, we look at the precise relation of thenoncrossing complex and the weak separability complex of Leclerc and Zelevinsky [Amer. Math.Soc. Transl. 181(2) (1998), 85–108]; see also Scott [J. Algebra 290(1) (2005), 204–220] amongothers. We show that the weak separability complex is not only a subcomplex of the noncrossing

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F. Santos, C. Stump and V. Welker 2

complex as noted by Petersen et al. [J. Algebra 324(5) (2010), 951–969] but actually its cyclicallyinvariant part.

2010 Mathematics Subject Classification: 52B20 (primary); 06A11 (secondary)

1. Introduction and main results

Let [n] denote the (ordered) set {1, . . . , n} of the first n positive integers. Twopairs (i < i ′) and ( j < j ′) with i 6 j are said to nest if i < j < j ′ < i ′ and crossif i < j < i ′ < j ′. In other words, they nest and cross if the two arcs nest and,respectively, cross in the following picture,

1 · · · i < j < j ′ < i ′ · · · n 1 · · · i < j < i ′ < j ′ · · · n.

Nestings and crossings have been intensively studied and generalized in theliterature, see for example [Ath98, PPS10, Pyl09, RS10]. One important contextin which they appear are two pure and flag simplicial complexes ∆NN

n and ∆NCn .

Recall that a flag simplicial complex is the complex of all vertex sets of cliquesof some graph.∆NN

n is the flag complex having the arcs 1 6 i < j 6 n as verticesand pairs of nonnesting arcs as edges, while ∆NC

n is the flag complex with thesame vertices and pairs of noncrossing arcs as edges.

It is not hard to see that the maximal faces of ∆NNn are parametrized by Dyck

paths of length 2(n − 2), while the maximal faces of ∆NCn are parametrized by

triangulations of a convex n-gon. Thus both complexes have the same number ofmaximal faces, the (n−2)nd Catalan number 1/(n − 1)

(2n−4n−2

). Moreover, it can be

shown that their face vectors coincide and that both are balls of dimension 2n− 4.In addition, the complex∆NC

n is the join of an (n−1)-dimensional simplex and anubiquitous (n − 4)-dimensional polytopal sphere ∆NC

n , the (dual) associahedron.

1.1. The nonnesting complex. The following generalization of the nonnest-ing complex is well known. Let Vk,n denote the set of all vectors (i1, . . . , ik),1 6 i1 < · · · < ik 6 n, of length k with entries in [n].

DEFINITION 1.1. Two vectors I = (i1, . . . , ik) and J = ( j1, . . . , jk) in Vk,n arenonnesting if for all indices a < b the arcs (ia < ib) and ( ja < jb) are nonnesting.The (multidimensional) nonnesting complex ∆NN

k,n is the flag simplicial complexwith vertices Vk,n and with edges being the nonnesting pairs of vertices.

By definition, we have ∆NN2,n = ∆NN

n . Equipped with the component-wiseorder, Vk,n becomes a distributive lattice. Moreover, I, J ∈ Vk,n are nonnesting





Noncrossing sets and a Grassmann associahedron 3

Figure 1. An order filter in the poset P3,7, and the corresponding monotone path inthe dual grid. Since the second, fourth and fifth steps on the path are going south,the corresponding element of V3,7 is (2, 4, 5).

if and only if I > J or J > I component-wise. That is, ∆NNk,n is the order complex

of the distributive lattice Vk,n .By Birkhoff’s representation theorem for distributive lattices [Bir37], there is

a poset P such that the distributive lattice Vk,n is isomorphic to the lattice of orderfilters of P . Remember that an order filter in P is a subset F ⊂ P satisfyinga ∈ F, a <P b⇒ b ∈ F. It is easy to see that, indeed, Vk,n is the lattice of filtersin the product poset Pk,n of a k-chain and an (n − k)-chain. A graphical way toset up this bijection between vectors in Vk,n and order filters in Pk,n is illustratedin Figure 1. To each filter in Pk,n associate a monotone lattice path from (0, 0) to(k, n − k) in a grid ‘dual’ to the Hasse diagram of Pk,n . The path is defined byseparating the elements in the filter from those not in the filter. Such paths bijectto Vk,n in the usual way by selecting the indices of steps in the direction of the firstcoordinate (the south direction in the picture). As long as there is no ambiguity,we thus consider elements of Vk,n as increasing k-tuples, as k-subsets, or as orderfilters in Pk,n .

By a result of Stanley [Sta86, Section 5], ∆NNk,n is the standard triangulation of

the order polytope Ok,n ⊆ [0, 1]k×(n−k) of Pk,n , where a vector I ∈ Vk,n is mappedto the characteristic vector χI ∈ NPk,n of the corresponding order filter. We referto Section 1.4.4 for basic facts about order polytopes and their triangulations.It follows that ∆NN

k,n is a simplicial ball of dimension k(n − k). Through thisconnection, its h-vector is linked to the Hilbert series of the coordinate ring ofthe Grassmannian Gk,n of k-planes in Cn . For details on this connection we referto Section 1.3.

Linear extensions of Pk,n , that is, maximal faces of ∆NNk,n , are in bijection with

standard tableaux of shape k × (n − k). Here, a tableau of shape k × (n − k) isa matrix in Nk×(n−k) that is weakly increasing along rows from left to right andalong columns from bottom to top. Equivalently, it is a weakly order preserving






Figure 2. Parts of the nonnesting complex ∆NN2,5 of the noncrossing complex ∆NC

2,5.

map Pk,n → N. We denote the set of all tableaux of this shape by Tk,n . A tableauis called standard if it contains every integer 1 through k(n − k) exactly once.An application of the hook length formula implies that maximal faces of ∆NN

k,n arecounted by the (n − k, k)th multidimensional Catalan number

Catn−k,k := 0!1! · · · (k − 1)!(n − 1)!(n − 2)! · · · (n − k)! (k(n − k))!.

These numbers were studied for example in [GP13, Sul04], see as well [Slo13,Seq. A060854]. Denote the h-vector of ∆NN

k,n by (h(k,n)0 , . . . , h(k,n)n(n−k)). It followsfrom the connection of ∆NN

k,n to the Hilbert series of the Grassmannian, and it wasalso observed in [Sul04] going back to MacMahon’s study of plane partitions,that its entries are the multidimensional Narayana numbers. We refer to [Sul04]for an explicit formula of these numbers, which can be combinatorially definedin terms of standard tableaux of shape k × (n − k) as follows. Call an integera ∈ [k(n− k)− 1] a peak of a standard tableau T if a+ 1 is placed in a lower rowthan a. Then, h(k,n)i equals the number of standard tableaux with exactly i peaks.This combinatorial interpretation implies in particular that

h(k,n)i ={

1 if i = k(n − k)− n + 10 if i > k(n − k)− n + 1

. (1)

EXAMPLE 1.2. For n = 5 and k = 2, the vertices of the nonnesting complex∆NN2,5

are given by V2,5 = {12, 13, 14, 15, 23, 24, 25, 34, 35, 45}, and the 5 maximalfaces are given by the join of the simplex spanned by {12, 13, 35, 45} and the 5faces

{{14, 15, 25}, {14, 24, 25}, {23, 24, 25}, {23, 24, 34}, {14, 24, 34}}.This subcomplex is shown in Figure 2 on the left.

1.2. The noncrossing complex. The reformulation of the nonnesting complexas the standard triangulation of Ok,n raises the question whether an analogous






construction of a multidimensional noncrossing complex has interestingproperties as well. The main object of study in this paper is the followingslight modification of Definition 1.1, introduced in [PPS10].

DEFINITION 1.3. Two vectors I = (i1, . . . , ik) and J = ( j1, . . . , jk) in Vk,n

are noncrossing if for all indices a < b with i` = j` for a < ` < b, the arcs(ia < ib) and ( ja < jb) do not cross. The (multidimensional) noncrossing complex∆NC

k,n is the flag simplicial complex with vertices Vk,n and with edges being thenoncrossing pairs of vertices.

REMARK 1.4. The definition in [PPS10] allows the vectors I and J to havedifferent lengths, and restricts to our definition in the case of equal lengths. Wediscuss this further in Section 1.4.1.

EXAMPLE 1.5. For n = 5 and k = 2, the vertices of the noncrossing complex∆NC

2,5 are again given by V2,5 = {12, 13, 14, 15, 23, 24, 25, 34, 35, 45}, and the 5maximal faces are given by the join of the simplex spanned by {12, 23, 34, 45, 15}and the 5 faces

{{14, 24}, {24, 25}, {13, 25}, {13, 35}, {14, 35}}.

The noncrossing complex is shown in Figure 2 on the right, where the circleindicates the simplex spanned by {12, 23, 34, 45, 15}.

REMARK 1.6. The reader may wonder why in the noncrossing world one requiresthe noncrossing property only for some pairs of coordinates a < b, while in thenonnesting world the nonnesting property is required for all pairs. One answeris that the direct noncrossing analogue of Definition 1.1 does not even yielda pure complex. But another answer is that it would not make a differencein Definition 1.1 to require the condition only for pairs with i` = j` fora < ` < b. All other pairs would automatically be nonnesting, thanks to thefollowing transitivity of nonnestingness: let a < b < c and suppose that the arcs(ia < ib) and ( ja < jb) are nonnesting, and the arcs (ib < ic) and ( jb < jc) arenonnesting as well. Then the arcs (ia < ic) and ( ja < jc) are also nonnesting.

The main properties of ∆NCk,n are summarized in the following statement.

THEOREM 1.7. The noncrossing complex ∆NCk,n is a flag, regular, unimodular and

Gorenstein triangulation of the order polytope Ok,n . In particular, ∆NCk,n and ∆NN

k,nhave the same f - and h-vectors.






This statement, of which we give an independent proof, is already containedin [PPS10] in the following way. There, the order polytope Ok,n appears asthe Gelfand–Tsetlin polytope of a particular shape (a rectangle). Theorem 8.1from [PPS10] says that ∆NC

k,n is a regular triangulation of it, and Theorem 8.7 thatit is Gorenstein. Unimodularity is mentioned in the proof of Corollary 8.2.

The claim that ∆NCk,n is ‘in some respects nicer’ than ∆NN

k,n is justified by theword ‘Gorenstein’ in the statement, which fails for ∆NN

k,n . Recall that a Gorensteintriangulation of a polytope is one that decomposes as the join of a simplex anda sphere (see Section 1.4.4 for details on such triangulations). This property isrelated to the last of the following list of purely combinatorial properties of ∆NC

k,n ,which generalize well known properties of the dual associahedron.

PROPOSITION 1.8. The complex ∆NCk,n has the following properties.

(i) The map a 7→ n + 1− a induces an automorphism on ∆NCk,n .

(ii) The map I 7→ [n]\I induces an isomorphism ∆NCk,n ˜−→∆NC

n−k,n .

(iii) I, J ∈ Vk,n are noncrossing if and only if they are noncrossing whenrestricting to the symmetric difference I4J = (I ∪ J )\(I ∩ J ).

(iv) For b ∈ [n] The restriction of∆NCk,n to vertices with b ∈ I yields∆NC

k−1,n−1. Therestriction of ∆NC

k,n to vertices with b 6∈ I yields ∆NCk,n−1.

(v) The n vertices in Vk,n obtained by cyclic rotations of the vertex (1, 2, . . . ,k) ∈ Vk,n do not cross any other vertex in Vk,n and hence are contained inevery maximal face of ∆NC

k,n .

Parts (ii) and (v) are mentioned in [PPS10, Remark 2.7] and [PPS10, Lemma8.6], where the n vertices in (v) are called solid elements.

Proof. Property (i) is clear from the definition. Property (ii) can be derived fromthe observation that a crossing between two vertices I, J ∈ Vk,n induces a crossingbetween [n]\I and [n]\J in Vn−k,n . Applying this argument twice, we obtainthat I and J are noncrossing if and only if [n]\I and [n]\J are noncrossing.To obtain Property (iii), observe that it is clear from the definition that one canalways restrict the attention to the situation where the set [n] is replaced by I ∪ J .It then follows with Property (ii) that one can as well remove I ∩ J . Property (iv)is a consequence of Property (iii). For Property (v), let J = ( j1, . . . , jk) and letI = (i1, . . . , ik) = (c, . . . , c + k) for some 1 6 c 6 n, with elements consideredmodulo n. Since the entries in I are as ‘close together’ as possible, it is notpossible to have two crossing arcs (ia < ib) and ( ja < jb) such that i` = j`for a < ` < b.






Observe that Properties (ii) and (iv) are natural when considering therelation between Ok,n and the Grassmannian (see Section 1.3) as they reflectthe isomorphism Gk,n

∼= Gn−k,n and the embeddings Gk−1,n−1 ↪→ Gk,n andGk,n−1 ↪→ Gk,n .

Properties (i), (ii), (iii), and (iv) also hold for the nonnesting complex ∆NNk,n ,

while Property (v) fails to hold. It implies that ∆NCk,n is the join of an (n − 1)-

dimensional simplex and a complex ∆NCk,n of dimension k(n − k) − n which

has the same h-vector as ∆NCk,n . Note that in particular, ∆NC

2,n is the (dual)associahedron ∆NC

n . Thus the following corollary (which implies that ∆NCk,n is

a Gorenstein triangulation of Ok,n) together with the discussion in Section 1.3justifies that we call the dual complex of ∆NC

k,n the Grassmann associahedron.

COROLLARY 1.9. ∆NCk,n is a flag polytopal sphere of dimension k(n − k) − n.

Moreover, Properties (i), (ii), and (iv) in Proposition 1.8 also hold for ∆NCk,n .

Observe that although [PPS10] show that ∆NCk,n is a sphere (Lemma 8.6 and

Theorem 8.7), polytopality of this sphere is a special case of their Conjecture8.10. The following arguments, applied to their Theorems 8.1 and 8.7 instead ofour Theorem 1.7 and Proposition 1.8(v), prove that conjecture in full generality.

Proof. ∆NCk,n is clearly a sphere or a ball of dimension k(n − k)− n, since it is the

link of an (n − 1)-simplex in the triangulation ∆NCk,n of the k(n − k)-dimensional

polytope Ok,n . Since hk,nk(n−k)−n+1 = 1 it must be a sphere. Flagness is preserved

under taking links and the three operations in the proposition are also preservedsince they leave the set of vertices described in (v) invariant.

Polytopality is an immediate consequence of regularity of ∆NCk,n , as we now

explain. ∆NCk,n being regular means that it can be realized as part of the boundary

of a certain convex polytope P . Let F be the (n − 1)-face of that polytopespanned by the vertices mentioned in Property (v) of Proposition 1.8. Then, ∆NC

k,n isgeometrically realized as the face figure of F in P , which is a polytope. (The facefigure of a simplicial face can be defined as the iterated vertex figure of all itsvertices.)

In particular, the Grassmann associahedron can be realized as a simple polytopeof dimension k(n − k)− n + 1 = (k − 1)(n − k − 1).

REMARK 1.10. Proposition 1.8(i) says that ∆NCk,n possesses the reflection

symmetry present in the associahedron ∆NCn . Of course, another symmetry

of ∆NCn comes from the cyclic rotation i 7→ i + 1 (considered as remainders

1, . . . , n modulo n). That symmetry does not carry over to ∆NCk,n for k > 3.






In fact, such a cyclic symmetry cannot carry over to the general situation sinceno flag complex on the set of vertices V3,6 that has the h-vector of ∆NN

3,6 can beinvariant under cyclic rotation. To see this, observe that such a complex wouldhave 155 edges and 35 nonedges. In particular, there should be in

(V3,62

)at least

two rotational orbits of size not a multiple of three (one orbit of edges and oneorbit of nonedges). But an orbit whose size is not divisible by three must have allits elements fixed by the order three rotation i 7→ i + 2, and the only element of(V3,6

2

)fixed by this rotation turns out to be {135, 246}.

1.3. Motivation: the Hilbert series of the Plucker embedding. Besides itswell behaved combinatorial properties, our main motivation for studying thenoncrossing complex comes from the connection between the order polytope Ok,n ,initial ideals of the ideal of Plucker relations, and Hilbert series of Grassmannians.We refer to [Stu96, GL96, Hib87] for more details of this connection.

Let Gk,n denote the Grassmannian of k-dimensional linear subspaces in Cn , andlet Lk,n be the defining ideal of Gk,n in its Plucker embedding. Thus, Lk,n is thehomogeneous ideal in the polynomial ring

Tk,n = C[xi1,...,ik : 1 6 i1 < · · · < ik 6 n]generated by the Plucker relations. It follows from work of Sturmfels [Stu96]that the Stanley–Reisner ideals of all regular unimodular triangulations of Ok,n aresquarefree initial ideals of Lk,n . Indeed, let Mk,n be the ideal in the polynomial ringwith variables {xF : F filter of Pk,n} generated by the binomials xE xF − xE∩F xE∪F

for all choices of order ideals E and F in Pk,n . The ideal Mk,n is known as theHibi ideal of the poset Pk,n , or the Ehrhart ideal of the polytope Ok,n . In [Stu96,Proposition 11.10, Corollary 8.9] it is shown that Mk,n appears as an initialideal of Lk,n . In turn, it follows from [Stu96, Ch. 8] that there is a one to onecorrespondence between regular unimodular triangulations of Ok,n and squarefreemonomial initial ideals of Mk,n . This correspondence sends a particular regularunimodular triangulation to its Stanley–Reisner ideal.

• The regular, unimodular, flag triangulation ∆NNk,n of Ok,n leads to a squarefree

monomial initial ideal of Mk,n studied by Hibi [Hib87].

• The regular, unimodular, flag triangulation ∆NCk,n provides a new initial ideal

with particularly nice properties and leads to new insight in the Hilbert seriesof the coordinate ring Ak,n = Tk,n/Lk,n .

From the relation between initial ideals and unimodular triangulations statedabove it follows that this Hilbert series is given by

HAk,n (t) = H(t)/(1− t)k(n−k)+1,






where H(t)= h(k,n)0 +h(k,n)1 t+· · ·+h(k,n)k(n−k)tk(n−k) is the h-polynomial of any regular

unimodular triangulation corresponding of Ok,n . In particular, its coefficients arethe multidimensional Narayana numbers.

In the following, let ∆ be a simplicial complex whose Stanley–Reisner idealI∆ appears as an initial ideal of Lk,n . Then the following properties are desirablefor ∆:

• It follows from (1) that there are at most n variables that do not appear in theset of generators of I∆. Equivalently, if ∆ decomposes into ∆ = 2V ∗∆′ where2V is the full simplex spanned by V , then #V 6 n. Thus the ‘most factorizable’complex ∆ should be a join over a simplex spanned by n vertices.

• The fact that Ak,n is Gorenstein should be reflected in ∆. Thus we desirethat ∆ is the join of a simplex with a triangulation of a (homology) sphereof the appropriate dimension or, even better, the boundary complex of asimplicial polytope, which would then deserve the name (dual) Grassmannassociahedron.

• Since Ak,n has a quadratic Grobner basis, it is Koszul. Hence, one could hopethat I∆ is generated by quadratic monomials, or, equivalently, that ∆ is flag.

• One could hope that ∆ reflects the duality between Gk,n and Gn−k,n , as well asthe embeddings Gk−1,n−1 ↪→ Gk,n and Gk,n−1 ↪→ Gk,n .

Theorem 1.7, Proposition 1.8, and Corollary 1.9 say that the noncrossingcomplex ∆ = ∆NC

k,n fulfils all these properties.Note that in this algebraic framework the result of Theorem 2.3 translates into

a statement about standard monomials in Tk,n/Mk,n (the Hibi ring of Pk,n , or theEhrhart ring of Ok,n). Let� be a term order for Tk,n and suppose the correspondinginitial ideal of Mk,n is squarefree (and monomial). Equivalently, by Sturmfels’results, the initial ideal comes from a unimodular triangulation of Ok,n . Tableauxof shape k × (n − k) are nothing but the integer points in dilations of Ok,n (seeLemma 4.1) and hence they index standard monomials with respect to �. Moreprecisely, tableaux in the r th dilation of Ok,n correspond to standard monomialsof degree r . (Observe that there is a certain ambiguity here. Since Ok,n containsthe origin, a point in its r th dilation lies also in the sth dilation, for any s > r .This reflects the fact that multiplying by the generator corresponding to the vertex(n − k + 1, . . . , n) of Ok,n has no effect in the tableaux and is the reason whyV ∗k,n = Vk,n\{(n − k + 1, . . . , n)} appears in Theorem 2.3 instead of Vk,n .) Byassumption the initial ideal of Mk,n with respect to � is squarefree. Hence there isa simplicial complex ∆ such that the initial ideal of Mk,n is the Stanley–Reisnerideal of ∆ and consequently standard monomials for � are the monomials whose






support is a face in∆. Since each standard monomial is in a unique way a productof the variables, which are in bijection to the vertices of Ok,n or to Vk,n , standardmonomials of degree r are identified with multisets of r elements from Vk,n

whose support lies ∆. Thus combinatorially we get an identification of tableauxand multisets. In this perspective Theorem 2.3 provides this identification for∆ = ∆NN

k,n and ∆ = ∆NCk,n and the corresponding term orders.

Since Mk,n is an initial ideal of Lk,n , standard monomials for � are alsostandard monomials of a Grobner basis of Lk,n , which links our results to standardmonomial theory (see [LR08]) for Schubert varieties. Among other aspects, thistheory deals with straightening rules for products of standard monomials in thecoordinate rings. For∆NN

k,n we are in the classical standard monomial theory of theGrassmann variety. It would be interesting to develop straightening laws for ournew set of standard monomials corresponding to ∆NC

k,n .

1.4. Relation to previous work.

1.4.1. Petersen–Pylyavskyy–Speyer’s noncrossing complex. Some of the mainresults from this paper were previously proved by Petersen et al. in [PPS10] ina more general context. Let Vn denote the set of all subsets of [n], which canbe thought of as the disjoint union of Vk,n for all k ∈ [0, n]. Petersen et al. thendefine a noncrossing relation among elements of Vn and consider, for each subsetL ⊂ [n], the flag complex m(nc)

L of noncrossing vectors whose length belongs to L .In particular, m(nc)

{k} is exactly equal to the noncrossing complex ∆NCk,n considered

in this paper.The main result of [PPS10], as was already mentioned, is the generalization

of Theorem 1.7 to arbitrary L , by changing the order polytope Ok,n to the moregeneral Gelfand–Tsetlin polytopes of shape L .

Our methods of proof, however, are different, and, as we think, of independentinterest. Our main innovation is the explicit relation of facets of the noncrossingcomplex and tableaux that we set up in Section 2. This has several algorithmicapplications, analogous to the driving rules in [PPS10]:(1) Theorem 2.3 (or, rather, its proof) contains a fast algorithm for point location

in ∆NCk,n: given a point T in the order polytope Ok,n , the algorithm outputs the

minimal face of ∆NCk,n containing T (the carrier of T ).

(2) The ‘pushing of bars’ procedure described in the proof of Proposition 2.10gives an efficient algorithm to construct the noncrossing complex ∆NC

k,n orthe star of any individual face in it. Efficient here means ‘polynomial in theoutput size’.

As a by-product of the second item above, we have a natural way to give directionsto the edges in the dual graph of the noncrossing complex. In Section 2.3 we






show that these directions make the graph acyclic which, in particular, allowsus to define a poset structure on the facets of ∆NC

k,n . We call this the Grassmann–Tamari poset since it generalizes the classical Tamari poset, and conjecture it tobe a lattice. (After a preprint version of this work was available, this conjecturewas proven by McConville [McC17].)

1.4.2. Pylyavskyy’s noncrossing tableaux. In [Pyl09], Pylyavskyy introducesand studies what he calls noncrossing tableaux, showing that they areequinumerous with standard tableaux, hence with facets of∆NC

k,n . The constructiontherein does not seem to be directly linked to the multidimensional noncrossingcomplex, as already noted in [PPS10]. For example, Pylyavskyy’s noncrossingtableau are not in general monotone along columns, while the tableaux that webiject to maximal faces of ∆NC

k,n in Section 2.1 are strictly monotone along rowsand columns.

1.4.3. Weakly separable sets. Closely related to our complex is the notion ofweakly separable subsets of [n], introduced by Leclerc and Zelevinsky in [LZ98]in the context of quasicommuting families of quantum Plucker coordinates.Restricted to subsets of the same size k, which is the case of interest to us,the definition is that two k-subsets X, Y ⊂ [n] are weakly separable if, whenconsidered as subsets of vertices in an n-gon, the convex hulls of X\Y andY\X are disjoint. The flag complex ∆Sep

k,n of weakly separable k-subsets of [n]was studied by Scott in [Sco05, Sco06], who conjectured that ∆Sep

k,n is pure ofdimension k(n − k), and that it is strongly connected (that is, its dual graph isconnected). Both conjectures were shown to hold by Oh et al. [OPS15], for thefirst see also Danilov et al. [DKK10, Proposition 5.9].

It is not hard to see that ∆Sepk,n is a subcomplex of ∆NC

k,n and it is trivial toobserve that ∆Sep

k,n is invariant under cyclic (or, more strongly, dihedral) symmetry.As we see in Section 5, it turns out that the weak separation graph is theintersection of all cyclic shifts of the noncrossingness graph. Since flagness ispreserved by intersection, the same happens for the complexes. We expect ourapproach to the noncrossing complex ∆NC

k,n to also shed further light on theweak separability complex ∆Sep

k,n . In particular, we hope to better understand theintriguing conjecture about the topology of ∆Sep

k,n and its generalizations that canbe found in a preliminary version [HH11] of [HH13].

1.4.4. Triangulations of order polytopes. Essential for most of our mainconclusions is the fact that ∆NN

k,n and ∆NCk,n are triangulations of an order polytope.

We recall some basic facts about order polytopes and then give relations to knownresults about triangulations of order polytopes or more general integer polytopes.






Let P be a finite poset. The order polytope of P , introduced by Stanley [Sta86],is given by

O(P) := {(xa)a∈P ∈ [0, 1]P : xa 6 xb for all a <P b}= conv{χF : F order filter of P},

where χF ∈ NP is the characteristic vector of the order filter F of P . Theorder polytope is a 0/1-polytope of dimension |P|. It has a somehow canonicaltriangulation ∆(P), see again [Sta86, Section 5], that we call the standardtriangulation. It is also sometimes called the staircase triangulation of O(P). Itcan be described in the following equivalent ways.

• Each of the |P|! monotone paths from (0, . . . , 0) to (1, . . . , 1) in the unit cube[0, 1]P defines a full-dimensional simplex. These simplices triangulate the cube,and the subset of them whose vertices lie in O(P) triangulate O(P).

• Each such monotone path is the Hasse diagram of a linear extension of P .Thus, ∆(P) is the subdivision of O(P) into the order polytopes of the linearextensions of P .

• Under the correspondence between vertices of O(P) and filters of P , linearextensions correspond to maximal containment chains of filters. That is, ∆(P)is the order complex of the lattice of filters of P , where the order complex ofa poset is the flag simplicial complex obtained from the comparability graphof P .

• Last but not least, the complex ∆(P) can be realized as the partition of O(P)obtained by slicing it by all the hyperplanes of the form {xa = xb}, a, b ∈ P .Of course, these hyperplanes only slice O(P) if a and b were incomparable,in which case the two sides of the hyperplane correspond to the two possiblerelative orders of a and b in a linear extension of P .

The third (and also the fourth) description of ∆(P) shows that it is a flagcomplex. Any of the first three shows that it is unimodular (all simplices haveeuclidean volume 1/|P|), the minimal possible volume of a full-dimensionallattice simplex in RP . Finally, the last description implies it to be regular.

In [RW05] Reiner and Welker construct, for every graded poset P of rank n, aregular unimodular triangulation Γ (P) of O(P) that decomposes as 2W ∗ Γ (P)for a simplex 2W with n vertices and a polytopal sphere Γ (P). Since thisis a Gorenstein simplicial complex we call it a Gorenstein triangulation. Theexistence of Gorenstein triangulations was later verified by Athanasiadis [Ath05]for a larger geometrically defined class of polytopes and then by Bruns andRomer [BR07] for the even larger class of all Gorenstein polytopes admitting






a regular unimodular triangulation. A Gorenstein polytope, here, is one whoseunimodular triangulations have a symmetric h-vector, and it was first shownin [Hib87] that an order polytope O(P) is Gorenstein if and only if P is graded.The survey article by [CHT06] puts the existence of Gorenstein triangulations inan algebraic perspective.

In particular, any of [RW05, Ath05, BR07] shows the existence of aregular, unimodular, Gorenstein triangulation of O(Pk,n). This implies thatthe multidimensional Narayana numbers are the face numbers of a simplicialpolytope, and thus satisfy all conditions of the g-theorem. It can be checkedthat the triangulation of [RW05] is not flag for Pk,n and neither the resultsfrom [Ath05] nor from [BR07] can guarantee flagness of the triangulation.The construction in [PPS10] and the present paper does. In particular, themultidimensional Narayana numbers satisfy all inequalities valid for h-vectorsof flag simplicial polytopes. This includes the positivity of the γ -vector and asa special case the Charney–Davis inequalities. Note, that the latter implicationare know to hold by [Bra04], where they are shown to hold for all triangulationsof order polytopes of graded posets. Also, it was pointed out by Athanasiadisto the authors of [RW05] that the Gorenstein triangulation of O2,n obtainedfrom their construction is not isomorphic to a dual associahedron. To our bestknowledge, neither the construction from [Ath05] nor from [BR07] can be usedto obtain such a triangulation. Thus, the ∆NC

k,n from [PPS10] studied in this paperappears to be more suited for a combinatorial analysis, and more closely relatedto Grassmannians, than these previous constructions.

2. Combinatorics of the noncrossing complex

This section is devoted to the combinatorics of the noncrossing complex andits close relationship with the combinatorics of the nonnesting complex. We studynonnesting and noncrossing decompositions of tableaux, which will later be themain tool in Section 4 to understand the geometry of these complexes. We thendeduce several further combinatorial properties of these complexes directly fromthe tableau decompositions. In the final part of this section, we define and studythe Grassmann–Tamari order on maximal faces of the noncrossing complex.

2.1. The nonnesting and noncrossing decompositions of a tableau. Asdefined in the introduction, a tableau of shape k×(n−k) is a matrix T ∈ Nk×(n−k)

that is weakly increasing along rows from left to right and along columns frombottom to top. Recall also that we denote the set of all tableaux of shape k×(n−k)by Tk,n . We still consider rows as labelled from top to bottom (that is, the top rowis the first row). This unusual choice makes tableaux of zeros and ones correspond






to vectors in Vk,n . For each weakly increasing vector (b1, . . . , bk) ∈ [0, n−k]k , thetableau having as its ath row ba zeroes followed by n − k − ba ones correspondsto the increasing vector I = (b1 + 1, . . . , bk + k) ∈ Vk,n via the bijection sendingI ∈ Vk,n to its characteristic vector χI ∈ NPk,n .

We now show how to go from a multiset of vectors in Vk,n to a tableau, andvice versa. The geometric interpretation of tableaux as integer points in the conespanned by the order polytope Ok,n as discussed in Section 4 (see in particularLemma 4.1) will then lead to a proof that the nonnesting and the noncrossingcomplexes triangulate Ok,n .

Let L be a multiset of ` vectors (i1 j , . . . , ik j) ∈ Vk,n (1 6 j 6 `). The summingtableau T = (tab) of the multiset L is the k × (n − k)-matrix

tab = #{ j ∈ [`] : iaj 6 b + a − 1}.

Note that if L = {I } is a single vector, then the summing tableau has only zeroesand ones and coincides with χI , the characteristic vector of a filter in Pk,n . Thefollowing lemma can be seen as a motivation for the definition of the summingtableau, and is a direct consequence thereof.

LEMMA 2.1. The summing tableau T of a multiset L of vectors in Vk,n equals

T =∑I∈L

χI ∈ NPk,n .

In particular, T is a weakly order preserving map from Pk,n to the nonnegativeintegers and thus a tableau in Tk,n .

It follows directly from this description of the summing tableau that the twomaps described in Proposition 1.8(i) and (ii) translate to natural actions onsumming tableaux; see also Figure 1.

COROLLARY 2.2. The action on ∆NCk,n induced by a 7→ n + 1 − a corresponds

to a 180◦ rotation of the summing tableau. The map from ∆NCk,n to ∆NC

n−k,n inducedby I 7→ [n]\I corresponds to transposing the summing tableau along the north-west-to-south-east diagonal.

It will be convenient in the following to represent an `-multiset L of vectorsin Vk,n as the (k×`)-table containing the vectors in L as columns, in lexicographicorder. For example, let n = 7 and k = 3, and consider the multiset given by the(3× 9)-table






3

2

1

1 2 3 4 5 6 7 8 9

1 1 1 2 2 2 2 3 5

2 2 3 3 4 4 5 5 6

3 4 5 5 5 5 7 7 7

L =

Its summing tableau is

3 7 8 8

2 4 6 8

1 2 6 6

T =

For example, the first row of T says that the first row of L contains three 1’s,four 2’s, and one 3, while the last vector (5, 6, 7) does not contribute to Tas χ(5,6,7) = 0 ∈ NP3,7 . As in this example, if L contains the vector I1 :=(n − k + 1, . . . , n) ∈ Vk,n , that vector does not contribute to the summing tableausince χI1

= 0 ∈ NPk,n . We thus set V ∗k,n = Vk,n\{(n − k + 1, . . . , n)} for laterconvenience.

The following statement is at the basis of our results about the two simplicialcomplexes ∆NN

k,n and ∆NCk,n .

THEOREM 2.3. Let T ∈ Tk,n . Then there is a unique multiset ϕNN(T ) and a uniquemultiset ϕNC(T ) of vectors in V ∗k,n whose summing tableaux are T , and such that

• the vectors in ϕNN(T ) are mutually nonnesting; and

• the vectors in ϕNC(T ) are mutually noncrossing.

In order to prove this, we provide two (almost identical) procedures to constructϕNN(T ) and ϕNC(T ).

Let T = (tab) ∈ Tk,n be a tableau, and let ` = max(T ) = t1,n−k be its maximalentry. We are going to fill a (k×`)-table whose columns give ϕNN(T ) and ϕNC(T ),respectively. Since we want each column to be in V ∗k,n , we have to fill the ath row(a ∈ {1, . . . , k}) with numbers in {a, . . . , a+n−k}. Moreover, in order to have Tas the summing tableau of the multiset of columns, the number a+b must appearin the ath row exactly ta,b+1− ta,b times, where we use the convention ta,0 = 0 andta,n−k+1 = max(T ). That is, we do not have a choice of which entries to use in eachrow, but only on where to put them. Our procedure is to fill the table row by rowfrom top to bottom, inserting the entries a + 1, . . . , a + n − k in increasing order(each of them the prescribed number of times) placing them one after the other






into the ‘next’ column in the ath row of the table. The only difference betweenϕNN and ϕNC is how the term ‘next’ is defined.

• To obtain ϕNN , ‘next’ is simply the next free box from left to right. In the aboveexample, the table gets filled as follows.

3

2

1

1 2 3 4 5 6 7 8

1 1 1 2 2 2 2 3

2 2 3 3 4 4 5 5

3 4 5 5 5 5 7 7

ϕNN(T ) =

1 2 3 4 5 6 7 8

1 2 3 4 5 6 7 8

1 2 3 4 5 6 7 8

To help the reader, in the top-left corner of each box we indicate the order inwhich a given entry is inserted into its row of the table. Also, we have markedwith a circle the last occurrence of each entry in each row. These circle marksare not needed for the proof of Theorem 2.3 but will become important later.

• To obtain ϕNC, ‘next’ is slightly more complicated. For two vectors v,w ∈ Nk

we say that v precedesw in revlex order if the rightmost entry ofw−v differentfrom 0 is positive. We chose the revlex-largest vector whose ath entry has notyet been inserted and for which the property of strictly increasing entries in acolumn is preserved. In other words, inserting an integer i into row a is done bylooking at the first a − 1 entries v = (v1, . . . , va−1) of all vectors that have notbeen assigned an ath entry yet and such that va−1 < i . Among those, we assign ito the revlex-largest free box, that is, to that v for which va−1 is maximal, thenva−2 is maximal, and so on. If this revlex-largest vector is not unique, we fillthe box of the left most of the choices, in order to maintain the table columns inlexicographic order. In the above example, the table now gets filled as follows.

3

2

1

1 2 3 4 5 6 7 8

1 1 1 2 2 2 2 3

2 2 5 3 3 4 5 4

3 5 7 4 5 5 7 5

ϕNC(T ) =

1 2 3 4 5 6 7 8

1 2 8 3 4 6 7 5

1 6 8 2 5 4 7 3

Observe that we could have described the choice of ‘next’ position in theprocedure ϕNN by saying that it means the revlex-smallest vector (in the samesense as above for ϕNC) for which the property of strictly increasing entriesin a column is preserved. This makes both procedures almost identical, onlyinterchanging revlex-smallest and revlex-largest in the choice of the box to insertthe next integer.






REMARK 2.4. Observe that these procedures could, by Proposition 1.8(i), alsobe applied ‘from bottom to top’ by first inserting the last row, and then filling thetable row by row by the analogous lex- and revlex-insertions.

DEFINITION 2.5. The multisets ϕNN and ϕNC obtained from a tableau T by theabove procedures are called the nonnesting decomposition and the noncrossingdecomposition of T .

Proof of Theorem 2.3. We start with proving that the procedures give what theyare supposed to: every two columns of ϕNN(T ) are nonnesting, and every twocolumns of ϕNC(T ) are noncrossing. To this end, let I = (i1, . . . , ik) andJ = ( j1, . . . , jk) be two such columns, and let a, b be two indices such that i` = j`for all ` such that a < ` < b. To show that if I and J in ϕNN(T ) (respectively inϕNC(T )) are nonnesting (respectively noncrossing), we have to show that ia < jaimplies ib 6 jb (respectively ib > jb). When assigning row b in the table, wesee ia, ia+1, . . . , ib−1 in the column containing I , and similarly ja, ja+1, . . . , jb−1

in the column containing J . In this situation, the column containing I is filledbefore the column containing J for ϕNN and after the column J for ϕNC. Thus,ib 6 jb for ϕNN and ib > jb for ϕNC.

To show uniqueness, suppose that we would have not chosen the revlex-smallest (respectively revlex-largest) column at some point in the procedure. Thesame argument as before then implies that we then would have created two nesting(respectively crossing) columns.

2.2. Further properties of the complexes∆NNk,n and∆NC

k,n. We emphasize thatTheorem 2.3 alone, suitably interpreted, implies that the complexes∆NN

k,n and∆NCk,n

are unimodular triangulations of the order polytope Ok,n . This interpretation iscarried out in Section 4, after some preliminaries on triangulations and orderpolytopes that we briefly survey in Section 3. Before getting there, we provein the remainder of this section further combinatorial properties of the twocomplexes ∆NN

k,n and ∆NCk,n . Some of these properties (for example the pureness

of the complexes in Corollary 2.12) follow also from the geometric results inthe subsequent sections, but we think that it is interesting to have independentcombinatorial proofs.

To further understand the combinatorics of ∆NNk,n and ∆NC

k,n , we mark (indicatedby a circle in the figures) the last occurrence of every nonmaximal integer placedin each row of ϕNN(T ) and of ϕNC(T ) in the above construction procedure. We callthem the marked positions. In symbols, for each a ∈ [k] and b ∈ [n− k] we markthe last occurrence of a + b − 1 that is placed in row a. Here, ‘last occurrence’






is meant in the order the integer is inserted into the given row. For two examples,see the above instances of ϕNN(T ) and of ϕNC(T ).

REMARK 2.6. One can ask in which way the marked positions differ if we fillthe table from bottom to top according to Remark 2.4. If the position of the lastoccurrence of the maximal integer a + n − k in row a is marked, it turns out thatboth procedures provide the same marked positions. In other words, the markedpositions do not depend on the procedure, but can be described purely in terms ofthe table L , except that the rule is different when describing the marked positionsof a noncrossing table or a nonnesting table. In both cases, we assume that thetable L has no repeated columns (if it has, only one copy carries marks). Thereis going to be one mark for each row a ∈ [k] and each b ∈ [n − k + 1]. The markwill be in one of the vectors in

La,b := {(i1, . . . , ik) ∈ L : ia = a + b − 1}.The rule to decide which vector carries the mark is:

• For the marks in the nonnesting table, the mark lies in the vector I =(i1, . . . , ik) ∈ La,b for which (i1, . . . , ia) is smallest and (ia+1, . . . , ik) issmallest. Observe that there is no inconsistency on what of the two rules welook at first, since nonnesting vectors are component-wise comparable. For thesame reason, ‘smallest’ means both lex-smallest and revlex-smallest.

• For the marks in the noncrossing table, the mark lies in the vector I =(i1, . . . , ik) ∈ La,b for which (i1, . . . , ia) is revlex-smallest and (ia+1, . . . , ik)

is lex-largest.

In the following three lemmas, we collect further properties of the tablesϕNN(T ) and ϕNC(T ), some of which can be detected using the information wherethe last occurrences of the entries are placed.

LEMMA 2.7. Let T ∈ Tk,n , and let L be either ϕNN(T ) or ϕNC(T ). We then havethe following properties for L.

(i) The columns in L are ordered lexicographically.

(ii) Tk,1 > 0 if and only if the vector (1, . . . , k) is a column of L.

(iii) Let L i and L i+1 be two consecutive columns of L. Then the first row in whichL i and L i+1 differ is equal to the first row in which L i has a marked position.






(iv) Two consecutive columns L i and L i+1 coincide if and only if L i has nomarked position.

Proof. These properties can be directly read off the insertion proceduresfor ϕNN(T ) and for ϕNC(T ) and the definition of the marked positions.

LEMMA 2.8. Let T ∈ Tk,n . We have the following property for ϕNN(T ) which doesnot hold for ϕNC(T ).

(i) Two consecutive columns of ϕNN(T ) differ in exactly those positions in whichthe left of the two has marked positions.

Proof. This is also a direct consequence of the procedure and the definition of themarked positions.

LEMMA 2.9. Let T ∈ Tk,n . We have the following properties for ϕNC(T ) that donot hold for ϕNN(T ).

(i) T is strictly increasing along rows if and only if all vectors of the form(b + 1, . . . , b + k) for b ∈ [n − k − 1] appear as columns in ϕNC(T ).

(ii) T is strictly increasing along columns if and only if all vectors of the form(1, . . . , a, n−k+a+1, . . . , n) for a ∈ [k−1] appear as columns in ϕNC(T ).

Proof. Observe that the revlex-max insertion ensures that the first insertion of agiven integer a + b − 1 into row a yields a partial vector (of length a) of theform (b, . . . , a + b − 1). This implies that for fixed b ∈ [n − k − 1], we havethat Ta,b < Ta,b+1 for all a ∈ [k] if and only if (b + 1, . . . , b + k) is a columnof ϕNC(T ), thus implying (i). A similar observation holds as well for the columnsof T . For fixed a ∈ [k−1], we have that Ta,b < Ta+1,b for all b ∈ [n−k] if and onlyif (1, . . . , a, n− k+ a+ 1, . . . , n) is a column of ϕNC(T ), thus implying (ii).

We are now ready to prove the following proposition from which we then derivethe pureness and the dimension of the nonnesting and the noncrossing complexes.

PROPOSITION 2.10. Let T ∈ Tk,n , and let L be either ϕNN(T ) or ϕNC(T ). If acolumn of L contains more than one marked position, then there exists a vectorin V ∗k,n that is not contained in L and which does not nest or cross, respectively,with any vector in L, depending on L being ϕNN(T ) or ϕNC(T ).

Proof. In the case of L = ϕNN(T ), this is a direct consequence of Lemma 2.8:Given a column with multiple marked positions, we can always insert a new






column to the right of this column where only the last marked position is changed.This vector is nonnesting with all other vectors by construction. For example, thesecond column in the above example for ϕNN(T ) has the second and the thirdposition marked. We can thus insert a new column between the second and thethird where only the last position is changed, thus being the vector (1, 2, 5).

The case of L = ϕNC(T ) is a little more delicate. First, we can assumethat Tk,1 > 0, and that T is strictly increasing along rows and columns. Otherwise,we can, according to Lemmas 2.7(ii) and 2.9, together with Proposition 1.8(v),always insert the missing cyclic intervals as columns into ϕNC(T ) and modify Taccordingly.

Given this situation and a column containing more than one marked position,one can ‘push’ marked position to obtain a new vector in V ∗k,n that is not yetcontained in L , and which is noncrossing with every column of L . Since it isenough for our purposes here, we describe the procedure of pushing the lastmarked position, similarly to the situation for ϕNN(T ). To this end, let b be thecolumn containing more than a single marked positions, and let the last markedposition be in row a. Moreover, let the value of this last marked position be x .Now, pretend that we had one more x to be inserted into row a in this table.The condition that T is strictly increasing along rows and columns implies thatit would indeed be possible to insert another x into row a. Let b′ be the columnin which this next x would be inserted. Then, it would be possible to add a newcolumn between columns b′ − 1 and b′ containing the vector given by the firsta − 1 entries of the previous column b′ together with the remaining entries ofcolumn b. Call the resulting table L ′. Observe that by construction, L ′ equals therevlex-max insertion table of its summing tableau. This implies that all columnsin L ′ are noncrossing. Moreover, the marked positions of L ′ are exactly thoseof L , except that the last marked position in the previous column b has moved tothe new column b′. As an example, we consider the following table (which is thepreviously considered table with all extra vectors inserted as described earlier inthis proof).

3

2

1

1 2 3 4 5 6 7 8 9 10 11

1 1 1 1 1 2 2 2 2 3 4

2 2 2 5 6 3 3 4 5 4 5

3 5 7 7 7 4 5 5 7 5 6

1 2 3 4 5 6 7 8 9 10 11

1 2 3 10 11 4 5 7 9 6 8

1 6 11 10 8 2 5 4 9 3 7

L =

Now, consider the last column b = 11, having marked positions in rows 1and 3, with values 4 and 6, respectively. So, a new last 6 would be inserted intocolumn b′ = 9. The resulting table L ′ has a new column between columns 8 and 9






consisting of the first 2 entries of the previous column 9 and the last entry of theprevious column 11, thus being the vector (2, 5, 6). We therefore get the followingtable, extending L by one column.

3

2

1

1 2 3 4 5 6 7 8 9 10 11 12

1 1 1 1 1 2 2 2 2 2 3 4

2 2 2 5 6 3 3 4 5 5 4 5

3 5 7 7 7 4 5 5 6 7 5 6

1 2 3 4 5 6 7 8 9 10 11 12

1 2 3 11 12 4 5 7 9 10 6 8

1 6 12 11 9 2 5 4 8 10 3 7

L ′ =

REMARK 2.11. Observe that a procedure similar to the pushing procedure ofthe last marked position can be used to push the first marked position. Moreconcretely, we have seen in Remark 2.6 that filling the table top to bottom orbottom to top produces the same marked positions. The first marked position in acolumn of the top to bottom procedure can thus be seen as the last marked positionof the same column of the bottom to top procedure. Thus, it can be pushed in theanalogous way as the last bar is pushed.

The following corollary is well known for∆NNk,n . For∆NC

k,n it can also be deducedfrom results in [PPS10].

COROLLARY 2.12. The simplicial complexes ∆NNk,n and ∆NC

k,n are pure ofdimension k(n − k).

Proof. Consider a face Fof ∆NNk,n not containing the vector (n − k + 1, . . . , n).

This is, F is a set of mutually nonnesting elements in V ∗k,n . As we haveseen in Theorem 2.3, F can be recovered from its summing tableau T , thatis, F = ϕNN(T ). Thus, we can recover the marked positions in the table asdescribed before. Alternatively, one can as well obtain the marked positions usingthe procedure described in Remark 2.6. Since the number of inserted markedpositions equals k(n − k) (assuming without loss of generality that T is strictlyincreasing along rows), we have |F | 6 k(n − k), as otherwise, we would haverepeated columns in F by Lemma 2.7(iv). Moreover, if |F | < k(n− k) then thereis a column containing more than one marked position. Thus, Proposition 2.10implies that there is a vector I ∈ V ∗k,n that is not contained in F such that F ∪ {I }is again mutually nonnesting and thus a face of ∆NN

k,n . This implies the corollaryfor ∆NN

k,n . The argument for ∆NCk,n is word by word the same.

REMARK 2.13. The proof of Corollary 2.12 contains an implicit characterizationof the tableaux that correspond to maximal faces of the nonnesting (respectively,






noncrossing) complex: they are those for which the nonnesting (respecti-vely, noncrossing) decompositions result in exactly one marked position ineach column of the table. For the nonnesting complex, in which all rows of thetable are filled from left to right, this condition is clearly equivalent to saying thatT contains each entry in [k(n − k)] exactly once. That is, ϕNN gives a bijectionbetween Standard Young Tableaux of shape k× (n− k) with max(T ) = k(n− k)and maximal faces of ∆NN

k,n .For∆NC

k,n we do not have a simple combinatorial characterization of the tableauxthat arise. According to Lemmas 2.7(ii) and 2.9, it is however easy to see that theymust be strictly increasing along rows and columns.

In the approach taken in [PPS10] the next two corollaries follow from theirTheorem 8.7.

COROLLARY 2.14. Every face of the reduced noncrossing complex ∆NCk,n of

codimension-1 is contained in exactly two maximal faces.

Proof. Let L be the table of a maximal face of ∆NCk,n , let b be the index of one

column of L , and let L ′ be the table of the codimension one face of ∆NCk,n obtained

from L by deleting a column b that does not contain a vector that is a cyclicrotation of the vector (1, . . . , k). Observe that it follows from Corollary 2.12that every pair L ′ ⊂ L of a codimension one face contained in a maximal faceis obtained this way. Now, every marked position in a column of L differentfrom column b is as well a marked position in L ′. Moreover, there is a uniquecolumn of L ′ that contains a unique second marked position. Following thepushing procedure described above, we obtain that pushing this marked positionagain yields the table L . Since the marked positions in L ′ are independentof L , and since we can push exactly those two marked positions in the uniquecolumn containing them, we conclude that L ′ is contained in exactly two maximalfaces.

Corollaries 2.12 and 2.14 together can be rephrased as follows.

COROLLARY 2.15. ∆NCk,n is a pseudomanifold without boundary. ∆NC

k,n is apseudomanifold with boundary, its boundary consisting of the codimension onefaces that do not use all the cyclic intervals.

2.3. The Grassmann–Tamari order. Based on our approach to the combi-natorics and on the geometry (developed in the next section) of the noncrossing






complex ∆NCk,n , we consider a natural generalization of the Tamari order on

triangulations of a convex polygon. To this end let the dual graph G(∆) of a puresimplicial complex ∆ be the graph whose vertices are the maximal faces of ∆,and where two maximal faces F1 and F2 share an edge if they intersect in a faceof codimension one.

We start with recalling the definition of the Tamari order. For furtherbackground and many more detailed, see for example [Rea12], and the TamariFestschrift containing that article. Fix n > 2. The elements of the Tamari poset Tn

are triangulations of a convex n-gon and the Hasse diagram of Tn coincides, as agraph, with the dual graph of∆NC

n = ∆NC2,n . To specify Tn we thus need to orient its

dual graph. Let T and T ′ be two triangulations which differ in a single diagonal.We then have that T4T ′ = {[i1, i2], [ j1, j2]}, and observe that (i1, i2) and ( j1, j2)

cross, so we can assume without loss of generality that i1 < j1 < i2 < j2. We saythat T ≺Tn T ′ form a cover relation in Tn if [i1, i2] ∈ T and [ j1, j2] ∈ T ′.

We next extend this ordering to the dual graph of∆NCk,n for general k. Let F be a

face of∆NCk,n of codimension one that uses all the cyclic intervals. Equivalently, by

Corollary 2.15, let F be a codimension one face that is not in the boundary of∆NCk,n .

Given the procedure we used in the proof of Proposition 2.10 we have that thetable of F contains exactly one column with more than a single marked position.This column contains exactly two marked positions. By Corollary 2.14, F iscontained in exactly two maximal faces, obtained by ‘pushing’ one of those twomarked positions.

DEFINITION 2.16. The Grassmann–Tamari digraph EG(∆NCk,n) is the orientation on

the dual graph G(∆NCk,n) given by the following rule. Let F1 and F2 be two maximal

faces sharing a codimension one face F . We orient the edge F1−F2 from F1 to F2

if F1 is obtained by pushing the lower of the two marked positions in the columnof F that has two marks. The Grassmann–Tamari order Tk,n is the partial orderon the maximal faces of ∆NC

k,n obtained as the transitive closure of EG(∆NCk,n). That

is, we have F1 <Tk,n F2 for two maximal faces if there is a directed path from F1

to F2 in EG(∆NCk,n).

Of course, the Grassmann–Tamari order will only be well defined if theGrassmann–Tamari digraph is acyclic (meaning that it does not contain directedcycles).

THEOREM 2.17. The Grassmann–Tamari digraph is acyclic, hence theGrassmann–Tamari order Tk,n is well defined. Moreover, Tk,n has a linearextension that is a shelling order of ∆NC

k,n .






Figure 3. The Grassmann–Tamari posets for ∆NC5,2 and for ∆NC

5,3.

Our proof of this theorem relies on the geometry of the noncrossing complex.It is thus postponed to Section 4.4.

EXAMPLE 2.18. Figure 3 shows the Grassmann–Tamari posets for n = 5,k ∈ {2, 3}.

PROPOSITION 2.19. The Grassmann–Tamari order has the following properties:

(1) The map on Tk,n induced by a 7→ n + 1− a is order reversing. In particular,Tk,n is self-dual.

(2) The map from Tk,n to Tn−k,n induced by I 7→ [n]\I is order reversing. Inparticular, Tk,n

∼= Tn−k,n .

Proof. Both parts of (1) follow from the discussion in Remarks 2.4, 2.6, and 2.11.For (2), let F1 and F2 be two maximal faces of∆NC

k,n sharing a face F = F1 ∩ F2

of codimension one, and let G i = [n]\Fi , i = 1, 2, and G = G1 ∩ G2 be the twocomplementary maximal faces in ∆NC

n−k,n and their intersection. Proposition 1.8(ii)implies that G is a face of codimension one as well. It is thus left to show thatif F1 is obtained from F by pushing the lower of the two marked positions in theappropriate column of the table for F , then G1 is obtained from G by pushingthe higher of the two marked positions again in the appropriate column of G.






To prove this, recall first that any column in any maximal face of ∆NCk,n and of

∆NCn−k,n (except the last) contains a unique marked position, and this column and

the column to its right coincide above this marked position by Lemma 2.7(iii).This implies that the values of the two marked positions in the table for F andin the table for G coincide. Since one of these two values is also the value of themarked position in F1, the other value must be the value of the marked positionin G1 (since G1 = [n]\F1). This finally yields that if pushing the lower entry in Fyields F1, then pushing the higher entry in G yields G1, as desired. As an example,consider the cover relation on the left in the two Grassmann–Tamari posets inFigure 3, and call the maximal faces F1 ≺T2,5 F2 in the left poset, and G2 ≺T3,5 G1

in the right poset. Then F = F1 ∩ F2 and of G = G1 ∩ G2 are given by and the

1 1 2 2 3 42 5 3 4 4 5F =

1 1 1 1 2 32 2 3 4 3 43 5 5 5 4 5

G =

values of the two marked positions in unique columns of F and G containing twomarked positions are 2 and 4. Pushing the 4 in F yields F1, and pushing the 2in G yields G1, as desired.

It is straightforward to see that the Grassmann–Tamari order restricts to theusual Tamari order for k = 2. The latter is well known to be a self-dual lattice.We conjecture this as well for the Grassmann–Tamari order for general k, testedfor n ∈ {6, 7, 8} and k = 3.

CONJECTURE 2.20. The Grassmann–Tamari poset Tk,n is a lattice.

This conjecture was recently proven by McConville in [McC17] in a moregeneral setting of Grid–Tamari posets. We refer to this paper for further details.

REMARK 2.21. It would be interesting to extend other properties of the (dual)associahedron or the Tamari lattice to ∆NC

k,n . An example of such a propertyconcerns the diameter. The diameter of the dual associahedron ∆NC

2,n is known tobe bounded by 2n − 10 for every n (Sleator et al. [STT88]) and equal to 2n − 10for n > 12 (Pournin, [Pou14]).

Although the proof of the 2n − 10 upper bound needs the use of the cyclicsymmetry of ∆NC

2,n (and, as we have seen in Remark 1.10, this symmetry does notcarry over to higher k), an almost tight bound of 2n − 6 can be derived fromthe very simple fact that every maximal face (triangulation of the n-gon) is atdistance at most n − 3 = (k − 1)(n − k − 1) from the minimal element in Tn .Unfortunately, the similar statement does not hold for Tk,n: For k = n − k = 3,






the above formula would predict that every maximal face can be flipped to theunique minimal element in Tk,n in 4 steps, but there are elements that need 5 suchflips. Moreover, there is no maximal face in ∆NC

3,6 that is connected to every othermaximal face by 4 or less flips.

Observe that Theorem 2.17 implies that the h-vector of the noncrossingcomplex ∆NC

k,n is the generating function of out-degrees in the Grassmann–Tamaridigraph. In particular,

h(k,n)i = |{T ∈ ∆NCk,n : T has i upper covers}|,

are the multidimensional Narayana numbers, and also equal to the number ofstandard Young tableaux with exactly i peaks; see Equation (1) in the Introductionand the preceding discussion. This raises the following problem.

OPEN PROBLEM 2.22. Is there an operation on peaks of standard Young tableaux(or, equivalently, on valleys of multidimensional Dyck paths) that

• describes the Grassmann–Tamari order directly on standard Young tableaux(or on multidimensional Dyck paths), thus also providing a bijection betweenmaximal faces of ∆NN

k,n and of ∆NCk,n , and

• which generalizes the Tamari order as defined on ordinary Dyck paths?

3. Order polytopes and their triangulations

3.1. Cubical faces in order polytopes. Let E and F be two order filters in aposet P , and let F(E, F) denote the minimal face of the order polytope O(P)containing the corresponding vertices χE and χF . It turns out that F(E, F) isalways (affinely equivalent to) a cube. Although this is not difficult to prove, itwas new to us and is useful in some parts of this paper.

To prove it, we start with the case when E and F are comparable filters. In thenext statement we use the notation EX for the segment going from the origin toX ∈ RP .

LEMMA 3.1. Let E ⊂ F be two comparable filters in a finite poset P and letP ′1, . . . , P ′d be the connected components of P|F\E . Then the minimal faceF(E, F) of the order polytope O(P) containing the vertices χE and χF is theMinkowski sum

χE + EχP ′1 + · · · + EχP ′d .

In particular, combinatorially F(E, F) is a cube of dimension d.






Proof. Observe that F(E, F) is contained in the face of O(P) obtained by settingthe coordinates of elements in E ∩ F = E to be 1 and those of elements not inE∪F = F to be 0. That face is just the order polytope of P|F\E (translated by thevector χE ). For the rest of the proof there is thus no loss of generality in assumingthat E = ∅ and F = P . We claim that the set of vertices of F(∅, P) is

{x = (xa)a∈P ∈ {0, 1}P : xa = xb if a and b are in the same component of P}.

To see that every vertex of F(∅, P)must have this form, observe that the equalityxa = xb for two comparable elements defines a face of the order polytope and itis satisfied both by χE and χF , so it is satisfied in all of F(∅, P). Moreover, sinceconnected components are the transitive closure of covering relations, the equalityxa = xb for two elements of the same connected component is also satisfiedin F(∅, P). For the converse, let x ∈ {0, 1}P be such that xa = xb when a and bare in the same component of P . Put differently, x is the sum of the characteristicvectors of some subset S of the components,

x =∑b∈S

χP ′b , for some S ⊂ [d].

Clearly, x is a vertex of O(P), since a union of connected components is a filter.Consider the complementary vertex

y =∑b 6∈S

χP ′b .

We have x + y = χ∅ + χP , which implies that the minimal face containing χ∅and χP contains also x and y (and vice versa).

The description of the vertices of F(∅, P) automatically implies theMinkowski sum expression for F(∅, P). Moreover, since the different segmentsEχP ′1, . . . , EχP ′d have disjoint supports, their Minkowski sum is a Cartesianproduct.

The last part of the proof has the following implications.

COROLLARY 3.2. Let E, F, E ′ and F ′ be four filters. Then the followingproperties are equivalent:

(1) F(E, F) = F(E ′, F ′).

(2) (χE ′, χF ′) is a pair of opposite vertices of the cube F(E, F).






(3) χE + χF = χE ′ + χF ′ .

(4) E ∩ F = E ′ ∩ F ′ and E ∪ F = E ′ ∪ F ′.

In particular, we have that F(E, F) = F(E ∩ F, E ∪ F) which, by the previouslemma, is combinatorially a cube.

3.2. Triangulations. Unimodularity, regularity, and flagness. Let Q be apolytope with vertex set V . A triangulation of Q is a simplicial complex ∆geometrically realized on V (by which we mean that V is the set of vertices of ∆,and that the vertices of every face of ∆ are affinely independent in Q) that coversQ without overlaps. A triangulation T of a polytope Q is called regular if there isa weight vector w : V → R such that T coincides with the lower envelope of thelifted point configuration

{(v,w(v)) : v ∈ V } ⊂ R|Q|+1.

See [DRS10] for a recent monograph on these concepts. Another way to expressthe notions of triangulations and regularity, more suited to our context, is asfollows.

• An abstract simplicial complex ∆ with its vertices identified with those of Qis a triangulation of Q if and only if for every x ∈ conv(Q) there is a uniqueconvex combination of vertices of some face σ of ∆ that produces x . That is,there is a unique σ ∈ ∆ (not necessarily full dimensional) such that

x =∑v∈σ

αvv

for strictly positive αv with∑

v αv = 1.

• ∆ is the regular triangulation of Q for a weight vector w : V → R if, for everyx ∈ Q, the expression of the previous statement is also the unique one thatminimizes the weighted sum

∑v∈V αvw(v), among all the convex combinations

giving x in terms of the vertices of Q.

Observe that for some choices ofw (most dramatically, whenw is constant) theweighted sum of coefficients may not always have a unique minimum. In this casew does not define a regular triangulation of Q but rather a regular subdivision. Itis still true that the support of every minimizing convex combination is containedin some cell of this regular subdivision, but it may perhaps not equal the set ofvertices of that cell.






If the vertices V of the polytope Q are contained in Zd (or, more generally, ina point lattice) we call a full-dimensional simplex unimodular when it is an affinelattice basis and we call a triangulation unimodular when all its full-dimensionalfaces are. All unimodular triangulations of a lattice polytope have the same f -vector and, hence, the same h-vector. See, for example, [DRS10, Section 9.3.3].This h-vector can be easily computed from the Ehrhart polynomial of Q and isusually called the Ehrhart h∗-vector of Q.

If we know a triangulation of a lattice polytope Q to be unimodular and flag,then checking its regularity is easier than in the general case. The minimality ofweighted sum of coefficients in convex combinations of points x ∈ Q needs to bechecked only for very specific choices of x and very specific combinations.

LEMMA 3.3. Let Q be a lattice polytope with vertices V , let ∆ be a flagunimodular triangulation of Q, and let w : V → R be a weight vector. Thenthe following two statements are equivalent:

(i) The complex ∆ is the regular triangulation corresponding to w,

(ii) For every edge v1v2 of the complex ∆ and for every pair of vertices {v′1, v′2}6= {v1, v2} with v1 + v2 = v′1 + v′2, we have w(v1)+w(v2) < w(v

′1)+w(v′2).

Proof. Let ∆w be the regular triangulation (or subdivision) produced by theweight vector w. Consider ∆w as a simplicial complex, even if it turns out not tobe a triangulation, taking as maximal faces the vertex sets of the full-dimensionalcells (simplices or not) of ∆w. We are going to show that this simplicial complexis contained in ∆. The containment cannot be strict because ∆w covers Q, so thiswill imply ∆ = ∆w.

Since ∆ is flag, to show that ∆w ⊂ ∆ it suffices to show that every edgeof ∆w is an edge in ∆. So, let v′1v

′2 be an edge of ∆w. For the rest of the

proof, we consider our polytope Q embedded in Rd × {1} ⊂ Rd+1. We regardthe vertices {v1, . . . , v`} of each unimodular simplex in ∆ as vectors spanning acone C , where spanning means not only linearly but also integrally (because ofunimodularity): every integral point in C is a nonnegative integral combinationof the vi ’s.

Consider then the point v′1 + v′2. It lies at height two in one of those cones,because v′1 + v′2 ∈ Rd × {2}. Thus, v′1 + v′2 = v1 + v2 for some edge v1v2 of∆. The hypothesis in the statement is then that either {v′1, v′2} = {v1, v2} (as weclaim) or w(v1)+w(v2) < w(v

′1)+w(v′2). The latter is impossible because then,

letting x = (v1+ v2)/2 = (v′1+ v′2)/2 be the midpoint of the edge v′1v′2 of∆w, the

inequality contradicts the fact that ∆w is the regular subdivision for w.






3.3. Central orientations of the dual graph and line shellings. Everyregular triangulation of a polytope is shellable. We sketch here a proof, adaptedfrom [DRS10, Section 9.5]. The ideas in it will be used in Section 4.4 for theproof of Theorem 2.17.

Let ∆ be a triangulation (regular or not) of a polytope Q, and let o be a pointin the interior of Q. We moreover assume o ∈ Q to be sufficiently generic so thatno hyperplane spanned by a codimension one face of ∆ contains o. We can thenorient the dual graph G(∆) ‘away from o’, in the following well-defined sense.Let σ1 and σ2 be two adjacent maximal faces in ∆ and let H be the hyperplanecontaining their common codimension one face. We orient the edge σ1σ2 of G(∆)from σ1 to σ2 if o lies on the same side of H as σ1. We call the digraph obtainedthis way the central orientation from o of G(∆) and denote it EG(∆, o).

LEMMA 3.4. If ∆ is regular then EG(∆, o) is acyclic for every (generic) o.Moreover, the directions in EG(∆, o) are induced by a shelling order in themaximal faces of ∆.

Proof. We are going to prove directly that there is a shelling order in ∆ thatinduces the orientations EG(∆, o). This implies EG(∆, o) to be acyclic.

The idea is the concept of a line shelling. One way to shell the boundarycomplex of a simplicial polytope Q is to consider a (generic) line ` going throughthe interior of Q and taking the maximal faces of Q in the order that the facet-defining hyperplanes intersect `. The line ` is considered to be closed (its twoends at infinity are glued together) and the maximal faces are numbered startingand ending with the two maximal faces intersecting `. Put differently, we canthink of the process as moving a point p = p(t), t ∈ [0, 1] along the line `,starting in the interior of Q, going through infinity, then back to the interior of Q,and recording the facets of Q in the order their facet-defining hyperplanes arecrossed by p(t). See Figure 4 (left) for an illustration and [Zie94, Ch. 8] formore details.

For a regular triangulation ∆, let ∆ be the convex hypersurface that projectsto ∆, and let ` be the vertical line through o. If we order the facets of ∆ (hence,the maximal faces of∆) in their line shelling order with respect to `, this orderingis clearly inducing the central orientation EG(∆, o) of G(∆). Moreover, it is ashelling order of the boundary of ∆ (and hence of ∆) because it is an initialsegment in the line shelling order of conv(∆) with respect to the line `. SeeFigure 4 (right) for an illustration.

REMARK 3.5. If the triangulation ∆ is not regular, then the central orientationEG(∆, o) may contain cycles.






Figure 4. The idea behind a line shelling, and the line shelling associated toa central orientation from a generic interior point o.

4. Geometry of the nonnesting and noncrossing complexes

The goal of this section is to show that the noncrossing complex ∆NCk,n is a

regular, unimodular, flag triangulation of the order polytope of the product oftwo chains. The method presented also yields the same result for the nonnestingcomplex ∆NN

k,n , which is well known.

4.1. The order polytope of the product of two chains. Recall from Section1.4.4 that the vertices of Ok,n (characteristic vectors of filters in Pk,n) are inbijection with the vertices of ∆NN

k,n (elements of Vk,n). As before, we use the samesymbol (typically I or J ) to denote an element of Vk,n and its associated filter.

To show that ∆NCk,n is a triangulation of Ok,n let us understand a bit more the

combinatorics of the maximal faces of Ok,n . These are of the following threetypes:

(1) There are two maximal faces corresponding to the unique minimal vectorI0 := (1, . . . , k) and the unique maximal vector I1 := (n + 1 − k, . . . , n).Each of these maximal faces contains all but one vertex, namely either χI0

=(1, . . . , 1) or χI1

= (0, . . . , 0). In particular, Ok,n is an iterated pyramid overthese two vertices.






(2) Each of the k(n−k−1) covering relations (a, b)l(a, b+1) in Pk,n producesa maximal face containing all χ(i1,...,ik ) with ik+1−a not equal to k + 1− a+ b.

(3) Each of the (k − 1)(n − k) covering relations (a, b)l (a + 1, b) produces amaximal face containing all χ(i1,...,ik ) not satisfying ik−a < k−a+b < ik+1−a .(That is, vectors in Vk,n not containing the entry k−a+b and having exactlya − 1 elements greater than k − a + b).

4.2. Tableaux as lattice points in the cone of Ok,n. Denote by O∗k,n themaximal face containing all vertices of Ok,n except the origin χI1

= (0, . . . , 0).Its vertex set is V ∗k,n = Vk,n\{(n − k + 1 · · · n)}, considered in Section 2. SinceOk,n is a pyramid over O∗k,n with apex at the origin, it is natural to study the coneCk,n over O∗k,n . That is,

Ck,n := R>0Ok,n = {λv ∈ RPk,n : λ ∈ [0,∞), v ∈ Ok,n}.

Equivalently, Ck,n is the polyhedron obtained from the inequality descriptionof Ok,n by removing the inequality xk,n−k 6 1.

Let us now look at the set Tk,n of all tableaux of shape k × (n − k). It is clearthat the inequalities describing weak increase are the same as those defining themaximal faces of the cone Ck,n . We hence have the following lemma.

LEMMA 4.1. Tk,n is the set of integer points in Ck,n .

Moreover, the summing tableau associated to a list of vectors I1, . . . , I` ∈ V ∗k,n isnothing but the sum of the characteristic vectors χI1, . . . , χI` of the correspondingvertices of Ok,n , see Lemma 2.1. With this in mind, Theorem 2.3 can be rewrittenas follows.

PROPOSITION 4.2. For every integer point T ∈ Ck,n there is a unique nonnegativeinteger combination of characteristic vectors χI for I ∈ V ∗k,n with noncrossingsupport that gives T , and another unique combination with nonnesting supportthat gives T .

When translated into geometric terms and identifying faces of ∆NNk,n and ∆NC

k,nwith the convex hulls of the corresponding characteristic vectors, this propositionhas the following consequence.

COROLLARY 4.3. The restrictions of ∆NNk,n and ∆NC

k,n to V ∗k,n are flag unimodulartriangulations of O∗k,n .






Proof. We provide the proof for ∆NCk,n . The claim for ∆NN

k,n follows in the sameway. We use the characterization of triangulations via uniqueness of the convexcombination of each x ∈ O∗k,n as a convex combination of vertices of a face in∆NC

k,n (see Section 3.2). Assume, to seek a contradiction, that there is an x ∈ O∗k,nthat admits two different combinations whose support is a face in ∆NC

k,n . That is,there are faces S1 and S2 in ∆NC

k,n and positive real vectors λ ∈ RS1 , µ ∈ RS2 , suchthat ∑

I∈S1

λIχI =∑I∈S2

µIχI .

Assume further that S1 and S2 are chosen minimizing |S1| + |S2| among the facesin ∆NC

k,n with this property. This implies that S1 and S2 are disjoint since commonvertices can be eliminated from one side of the equality, and that conv{χI : I ∈ S1}and conv{χI : I ∈ S2} intersect in a single point. This, in turn, implies that thispoint, and the vectors λ andµ, are rational. Multiplying them by suitable constantswe consider them integral. But then the tableau

T :=∑I∈S1

λIχI =∑I∈S2

µIχI

turns out to have two different noncrossing decompositions, contradictingProposition 4.2.

Since Ok,n is a pyramid over χ(n−k+1,...,n) = (0, . . . , 0) and (n − k + 1, . . . , n)is noncrossing and nonnesting with every element of Vk,n , Corollary 4.3 impliesthat both are unimodular triangulations of Ok,n .

THEOREM 4.4. ∆NNk,n and ∆NC

k,n are flag unimodular triangulations of Ok,n .

The theorem follows from Stanley’s work in [Sta86] for∆NNk,n and from [PPS10,

Corollary 7.2] together with a mention of unimodularity in the proof of [PPS10,Corollary 8.2] for ∆NC

k,n .

4.3. ∆NCk,n as a regular triangulation of Ok,n. For real parameters α1, . . . ,

αk−1 consider the following weight function on the set of vertices Vk,n of Ok,n .For each I = (i1, . . . , ik) ∈ Vk,n let

w(I ) = w(i1, . . . , ik) :=∑

16a<b6k

αb−aiaib.

We assume that the values α1, . . . , αk−1 are positive and we require that αi+1� αi .






LEMMA 4.5. Let I, J and X, Y be two different pairs of elements of Vk,n . If I andJ are noncrossing and χI + χJ = χX + χY then X and Y are crossing and

w(I )+ w(J ) < w(X)+ w(Y ).

Proof. We set I = (i1, . . . , ik), J = ( j1, . . . , jk), X = (x1, . . . , xk) and Y = (y1,

. . . , yk). Observe that χI + χJ = χX + χY implies {ia, ja} = {xa, ya} for everya ∈ [k]. Since the pairs are different it follows that X and Y cross. For each paira, b ∈ [k] we then have two possibilities:

• {{xa, xb}, {ya, yb}} = {{ia, ib}, { ja, jb}}. We then say that (X, Y ) is consistentwith (I, J ) on the coordinates a and b.

• {{xa, xb}, {ya, yb}} = {{ia, jb}, { ja, ib}}. We then say that (X, Y ) is inconsistentwith (I, J ) on the coordinates a and b.

Observe that the difference w(X)+ w(Y )− w(I )− w(J ) equals∑a,b

αb−a(ia jb + jaib − iaib − ja jb) = −∑a,b

αb−a(ia − ja)(ib − jb),

where the sum runs over all inconsistent pairs of coordinates with 1 6 a < b 6 k.Observe also that, by the choice of parameters αb−a the sign of this expressiondepends only on the inconsistent pairs that minimize b − a. We claim that allsuch ‘minimal distance inconsistent pairs’ have the property that i` = j` for alla < ` < b. This follows from the fact that if a < c < b and ic 6= jc then (a, b)being inconsistent implies that one of (a, b) and (b, c) is also inconsistent.

Then, the fact that I and J are not crossing implies that, for all such a and b,we have that

ia < ja < jb < ib or ja < ia < ib < jb.

In any case, (ia − ja)(ib − jb) < 0, so that w(X)+ w(Y ) > w(I )+ w(J ).

The regularity assertion of the following corollary was proved in [PPS10,Theorem 8.1].

COROLLARY 4.6. ∆NCk,n is the regular triangulation of Ok,n induced by the weight

vector w.

Proof. This follows from Lemmas 3.3 and 4.5.






REMARK 4.7. The same ideas show that the nonnesting complex is the regulartriangulation of Ok,n produced by the opposite weight vector −w. The onlydifference in the proof is that at the end, since (I, J ) is now the nonnesting pair,we have that

ia < ja < ib < jb or ja < ia < jb < ib,

so that (ia − ja)(ib − jb) > 0 and w(X)+ w(Y ) < w(I )+ w(J ), as needed.

This means that ∆NNk,n and ∆NC

k,n are in a sense ‘opposite’ regular triangulations,although this should not be taken too literally. What we claim for this particularwand its opposite −w may not be true for other weight vectors w producing thetriangulation ∆NC

k,n . Anyway, since ∆NNk,n is the pulling triangulation of Ok,n with

respect to any of a family of orderings of the vertices (any ordering compatiblewith comparability of filters), this raises the question whether ∆NC

k,n is the pushingtriangulation for the same orderings. See [DRS10] for more on pushing andpulling triangulations.

4.4. Codimension one faces of ∆NCk,n, and the Grassmann–Tamari order.

Here we prove Theorem 2.17; that is, that the Grassmann–Tamari order is welldefined, and that any of its linear extensions is a shelling order for ∆NC

k,n .The key idea is the use a central orientation of the dual graph, as introduced

in Section 3.3, by explicitly describing the hyperplane containing interiorcodimension one faces in the triangulation of Ok,n by ∆NC

k,n .First we define the bending vector b(I,X) ∈ RPk,n of a segment X = (ia1, . . . , ia2)

(where 1 6 a1 < a2 6 k) of an element I = (i1, . . . , ik) ∈ Vk,n . We assumeia2 < a2 + (n − k). That is, X does not meet the east boundary of the grid.This technical condition is related to the convention that, when marking the lastoccurrence of each inter in a row of a noncrossing table, we omit the mark for themaximal integer a + n − k in row a; compare Remark 2.6.

The definition heavily relies on looking at I as a monotone path in the dualgrid of Pk,n as explained in Figure 1, and X as a connected subpath in it, startingand ending with a vertical step. See Figure 5 where X is represented as a thicksubpath in a longer monotone path. Along X we have marked certain parts with a-dot or a ⊕-dot. Namely when traversing the path from top to bottom:

• the first and the last step in X (both vertical) are marked with a -dot and witha ⊕-dot, respectively, and

• the corners X turns left or right are also marked with a⊕-dot and with a-dot,respectively.






Figure 5. A segment of a path in the dual grid of P10,18 and the correspondingbending vector in RPk,n .

Observe that the dots alternate between ⊕ and , and that there is the samenumber of both along X .

The bending vector b(I,X) is the vector in {−1, 0,+1}Pk,n obtained as indicatedon the right side of Figure 5. We start with the zero vector {0}Pk,n . Every time Xbends to the right or to the left, we add a (+1,−1)-pair to the squares north-eastand south-west to the corner, with the −1 added to the square inside and the +1added to square outside the corner. Equivalently, the north-east square gets a +1added in left turns (⊕-corners) and a −1 added in right turns or ( corners) andvice versa for the south-west square. Additionally, we add +1 to the square westof the first step and −1 to the square east of the first step and the other way roundfor the last step. Observe that if the first (or last) vertical step in X directly ends ina corner, then there is a cancellation of a +1 and a −1, see Example 4.10 below.

Let now J = ( j1, . . . , jk) ∈ Vk,n be another vector, thought of as anothermonotone path in the dual grid of Pk,n . Then the scalar product of the bendingvector b(I,X) and the characteristic vector χJ is, by construction, given by thenumber of⊕-dots minus the number of-dots of the path X that J goes through.Indeed, a (+1,−1)-pair in the bending vector b(I,X) contributes to the scalarproduct if and only if J goes through the corresponding dot, and with the statedsign. In particular, the scalar product depends only on the intersection of the pathscorresponding to X and to J . The next lemma explains how to compute the scalarproduct by summing over the contributions from the connected components ofthis intersection. Observe that a connected component may be a single point.






LEMMA 4.8. Let I, J ∈ Vk,n be noncrossing and let X = (ia1, . . . , ia2) be asegment in I . Considering the connected components of the intersection of thepaths corresponding to X and to J , we have that

(1) a component contributes −1 to 〈b(I,X), χJ 〉 if and only if this componentstarts with ja1 = ia1 and this component of J leaves X to the left,

(2) a component contributes 1 to 〈b(I,X), χJ 〉 if and only if this component endswith ja2 = ia2 and this component of J enters X from the right,

(3) all other components contribute zero to 〈b(I,X), χJ 〉.

Proof. First, consider a component that passes neither through the initial -dotnor through the final ⊕-dot. Since I and J are noncrossing, J enters and exits Xon opposite sides. It therefore passes through equally many ⊕- and -dots, so itscontribution is zero. Next, consider a component that passes through the initial-dot. Equivalently, this component starts with ja1 = ia1 . Then there are twopossibilities. Either this component of J leaves X to the left, in which case itcontains one more -dot than ⊕-dots, that is, it contributes −1 to 〈b(I,X), χJ 〉.Or this component of J leaves X to the right and then this component containsas many -dots as it contains ⊕-dots, that is, it does not contribute to 〈b(I,X),χJ 〉. The argument for a component that passes through the final ⊕-dot, isanalogous.

Let F be an interior face of codimension one. Represent F by a table with twomarked positions in a certain column I = (i1, . . . , ik) and exactly one markedposition in the others, as in the proofs of Proposition 2.10 and of Corollary 2.14.Let a1 < a2 be the rows containing the two marked positions in I , and let F1

and F2 be the two maximal faces obtained from F by pushing the marked positionin the a1th row and in the a2th row, respectively. We then have the followinglemma.

LEMMA 4.9. The vector orthogonal to the hyperplane containing F is thebending vector b(I,X) where X = (ia1, . . . , ia2) is the segment of I startingwith the first marked position and ending with the second (and last) markedposition.

Moreover, F2 is on the same side of that hyperplane as b(I,X), while F1 is on theopposite side.

Proof. Let J be another column in the table for F . Since I and J are noncrossingwe can use Lemma 4.8 to compute 〈b(I,X), χJ 〉. The result is zero since theintersection of the paths corresponding to X and to J cannot contain a component






as described in Lemma 4.8(1) or (2), as a direct consequence of the descriptionof the marked positions given at the end of Remark 2.6. Observe here that ifthere was a component as described in Lemma 4.8(1), then ia1 = ja1 , and (ia1+1,

. . . , ik)would be lexicographically smaller than ( ja1+1, . . . , jk), contradicting thatdescription. Similarly, if there was a component as described in Lemma 4.8(2),then ia2 = ja2 , and (i1, . . . , ia2−1) is larger than ( j1, . . . , ja2−1) in reverselexicographic order, again contradicting that description.

Moreover, the same argument shows that if J1 and J2 are the new elementsin F1 and in F2 that were obtained by the pushing procedure, then the scalarproduct of b(I,X) and F1 is negative, while the scalar product of b(I,X) and F2 ispositive.

EXAMPLE 4.10. Consider the following summing tableau.

5 8 9 11

3 4 8 9

1 3 5 7

T =

Its associated noncrossing table, corresponding to a codimension one face in∆NC3,7,

is given as follows.

1 1 1 1 1 2 2 2 3 4 4

2 2 2 4 6 3 4 4 4 5 6

3 4 7 7 7 4 5 6 5 6 7

F =

The doubly marked column is I = (2, 4, 6), with X = (2, 4, 6) as well. Butobserve that, as monotone paths, X is a strict subpath of I since the initial andfinal horizontal steps in I are not part of X . The bending vector of X equals

1 0 -1 0

-1 1 1 -10 -1 0 1

b(I,X) =

Observe that two of the zeroes are obtained by cancelling a +1 and a −1 in thedefinition of the bending vector).

This is indeed orthogonal to the 11 tableaux corresponding to the columns ofF , which are given as follows.






1 1 1 1

1 1 1 11 1 1 1

1

2

3

1 1 1 1

1 1 1 10 1 1 1

1

2

4

1 1 1 1

1 1 1 10 0 0 0

1

2

7

1 1 1 1

0 0 1 10 0 0 0

1

4

7

1 1 1 1

0 0 0 00 0 0 0

1

6

7

0 1 1 1

0 1 1 10 1 1 1

2

3

4

0 1 1 1

0 0 1 10 0 1 1

2

4

5

0 1 1 1

0 0 1 10 0 0 1

2

4

6

0 0 1 1

0 0 1 10 0 1 1

3

4

5

0 0 0 1

0 0 0 10 0 0 1

4

5

6

0 0 0 1

0 0 0 00 0 0 0

4

6

7

Let us also compute the maximal faces F1 and F2 obtained pushing the 2 andthe 6 in the column corresponding to I in F .

1 1 1 1 1 2 2 2 2 3 4 4

2 2 2 4 6 3 4 4 4 4 5 6

3 4 7 7 7 4 5 6 7 5 6 7

F1 =

and

1 1 1 1 1 1 2 2 2 3 4 4

2 2 2 4 4 6 3 4 4 4 5 6

3 4 7 6 7 7 4 5 6 5 6 7

F2 =

The new elements J1 and J2 and their characteristic vectors χJ1 and χJ2 in thesemaximal faces are






J1 =0 1 1 1

0 0 1 10 0 0 0

2

4

7

J2 =1 1 1 1

0 0 1 10 0 0 1

1

4

6

.

As predicted by Lemma 4.9, χJ1 has a negative scalar product with b(I, X)and χJ2 has a positive scalar product.

In order to prove Theorem 2.17 we need to use one more property of bendingvectors:

LEMMA 4.11. Let o ∈ RPk,n−k be any vector such that

oa1,b1 + oa2,b2 > oa1,b2 + oa2,b1, for all 1 6 a1 < a2 6 k, 1 6 b1 < b2 6 n − k.

Then, for every I ∈ Vk,n and every segment X in I we have

〈b(I, X), o〉 > 0.

In particular, this holds for the vector o defined as oa,b = ab for every a, b ∈Pk,n−k .

Proof. The following additivity property follows trivially form the definition ofbending vectors: if X = (ia1, . . . , ia2) is a segment, and we decompose it into twoparts X1 = (ia1, . . . , ia) and X2 = (ia, . . . , ia2) via a certain a1 < a < a2, then

b(I, X) = b(I, X1)+ b(I, X2).

Via this property, we only need to prove the lemma for segments with only twoentries.

So, let X = (ia, ia+1) be such a segment. Its bending vector has exactly fournonzero entries. It has +1 in positions (a, ia − a) and (a+ 1, ia+1− a), and it has−1 in (a + 1, ia − a) and (a, ia+1 − a). By choice of o, 〈b(I, X), o〉 > 0.

Putting together Lemmas 3.4, 4.9 and 4.11 we can now easily proveTheorem 2.17.

Proof of Theorem 2.17. By Lemmas 4.9 and 4.11, the Grassmann–Tamariorientation in the dual graph of ∆NC

k,n coincides with the central orientationinduced by any o ∈ RPk,n−k satisfying the assumptions of Lemma 4.11. ByLemma 3.4 this orientation is acyclic and compatible with a shelling order of themaximal faces in ∆NC

k,n .






4.5. The nonnesting and noncrossing triangulations of the cube. We sawin Section 3.1 that every order polytope has some special faces that are cubes,namely the minimal face containing two given vertices. Here we describe thesefaces for Ok,n and study the triangulations of them induced by ∆NN

k,n and ∆NCk,n .

Let χI and χJ be two vertices of Ok,n . Remember from Section 3.1 thatthe minimal face of Ok,n containing χI and χJ has the following description.Let P1, . . . , Pd be the connected components of the poset given by the symmetricdifference of the filters corresponding to I and J . Then F(I, J ) is (affinelyequivalent to) a cube of dimension d whose vertices are given as

χI∩J + α1χP1 + · · · + αdχPd

for the 2d choices of a vector α = (α1, . . . , αd) ∈ {0, 1}d . In particular, χI and χJ

themselves correspond to certain antipodal vectors αI , αJ ∈ {0, 1}d .Our next observation is that the components P1, . . . , Pd come with a natural

linear order (and we consider them labelled according to that order). Indeed, ifwe think of I and J as monotone paths in the k × (n − k) grid, the componentsare the regions that arise between the two paths. Since the paths are monotone,we consider those regions ordered from left to right (or, equivalently, from top tobottom).

EXAMPLE 4.12. Consider the two vectors

I = (1, 2, 3, 5, 11, 12, 13, 14, 15, 21, 24),J = (3, 4, 5, 6, 7, 9, 12, 16, 18, 19, 20)

for k = 11, n = 24 with paths shown in Figure 6, compare also Figure 1. Theposet given by symmetric difference has the shown four connected componentslabelled 1 through 4 from left to right. Thus, F(I, J ) is combinatorially a 4-dimensional cube.

With this point of view, the following lemma is straightforward.

LEMMA 4.13. Let χI and χJ be two vertices of Ok,n , let d be the number ofconnected components of the symmetric difference of the filters corresponding toI and J , and let αI , αJ ∈ {0, 1}d be the 0/1-vectors identifying χI and χJ asvertices of F(I, J ). Then

• I, J are nonnesting if and only if {αI , αJ } = {(0, . . . , 0), (1, . . . , 1)}.• I, J are noncrossing if and only if {αI , αJ } = {(0, 1, 0, . . .), (1, 0, 1, . . .)}.

Lemma 4.13 has the following consequence.






Figure 6. Two paths in the dual grid of P11,24 and the corresponding fourcomponents.

COROLLARY 4.14. With the same notation, let χX and χY be two vertices ofF(I, J ), and let αX , αY ∈ {0, 1}d be the 0/1-vectors identifying them as verticesof F(I, J ). Then

• X, Y are nonnesting if and only if one of αX and αY is coordinatewise smallerthan the other.

• X, Y are noncrossing if and only if, αX and αY alternate between 0 and 1 in thecoordinates in which they differ.

This shows that if we restrict the triangulations induced by∆NNk,n and∆NC

k,n to thecube F(I, J ) they coincide with the following well known triangulations of thiscube.

• ∆NNk,n induces the standard triangulation of the cube, understood as the order

polytope of an antichain. That is, it is obtained by slicing the cube along allhyperplanes of the form xi = x j .

• ∆NCk,n induces a flag triangulation whose edges are the 0/1-vectors that alternate

relative to one another. It can also be described as the triangulation obtainedslicing the cube by all the hyperplanes of the form

xi + · · · + x j = z






for every pair of coordinates 1 6 i < j 6 d and for every z ∈ [ j − i]. We callit the noncrossing triangulation of the cube.

REMARK 4.15. The triangulation of the cube induced by ∆NCk,n was first

constructed by Stanley [Sta77] and then (as a triangulation of each of thehypersimplices) by Sturmfels [Stu96]. Lam and Postnikov [LP07] showed thetwo constructions to coincide.

Both the triangulation induced by ∆NNk,n and ∆NC

k,n are images of the dicingtriangulation of an alcoved polytope for the root system Ad . Alternatively, onecan say they are (after a linear transformation) Delaunay triangulations of afundamental parallelepiped in the lattice A∗d . Dicing triangulations of alcovedpolytopes in type A are always regular, unimodular and flag [LP07].

Note that the nonnesting triangulation of Ok,n is also induced by hyperplanecuts, but the noncrossing triangulation is not. This can be seen, for example, in∆NC

2,5, see Figure 2 on page 4.

REMARK 4.16. The (dual graph) diameters of the standard and the noncrossingtriangulations of the cube F(I, J ) are easy to compute from the fact that theyare obtained by hyperplane cuts: the distance between two given maximal facesequals the number of cutting hyperplanes that separate them.

In the standard triangulation every maximal face is separated from its oppositeone by all the

(d2

)hyperplanes, so the diameter equals

(d2

). For the noncrossing

triangulation a slightly more complicated argument gives that the diameter is(d+13

). In both bounds, d 6 min{k, n − k} is the dimension of the cubical face

F(I, J ).

4.6. Ok,n as a Cayley polytope. Let ∆` = conv{v1, . . . , v`} be an (` − 1)-dimensional unimodular simplex and let Q1, . . . , Q` be lattice polytopes in Rk .We do not require the individual Qi ’s to be full dimensional, but we require it fortheir Minkowski sum. The Cayley sum or Cayley embedding of the Qi ’s is the(k + `)-dimensional polytope

C(Q1, . . . , Qk) := conv{ k⋃

i=1

Qi × {vi}}⊂ Rk+`.

We show that for each of the k rows (or for each of the n−k columns) of the posetPk,n we can derive a representation of Ok,n as a Cayley sum. Let a ∈ [k] be fixed,and for each b ∈ [0, . . . , n − k] let V a,b

k,n be the set of vectors from Vk,n that havean a + b in their ath entry. In terms of tableaux, the vectors in V a,b

k,n correspond tothose tableaux that have a 0 in position(a, b) and a 1 in position (a, b+1). Denoteby Oa,b

k,n the convex hull of V a,bk,n . We then have the following lemma.






LEMMA 4.17. For every a ∈ [k],

Ok,n = C(Oa,0k,n , . . . ,Oa,n−k

k,n ).

This has enumerative consequences for the numbers of nonnesting (that is,standard Young) and noncrossing tableaux.

DEFINITION 4.18. Let Q1, . . . , Q` be an `-tuple of polytopes in Rk . For eachm = (m1, . . . ,m`) ∈ N` with

∑m i = k (equivalently, for each monomial

of degree k in R[x1, . . . , x`]) call the coefficient of xm in the homogeneouspolynomial

vol(x1 Q1 + · · · + x`Q`),

the m-mixed volume of Q1, . . . , Q`. Here the volume is meant normalized to thelattice. That is, unimodular simplices are considered to have volume 1.

We need the following consequence of the Cayley trick as given in [DRS10,Theorem 9.2.18].

LEMMA 4.19. Let Q1, . . . , Q` ⊆ Rk be an `-tuple of polytopes and let ∆ be aunimodular triangulation of C(Q1, . . . , Q`). Then, for each tuple m = (m1, . . . ,

m`) ∈ N` of sum k, the m-mixed volume of the tuple equals the number of maximalfaces of ∆ that have exactly mb + 1 vertices in each fibre {vb} × Qb.

Applied to the representation of Ok,n as a Cayley sum from Lemma 4.17 andtaking into account that both ∆NN

k,n and ∆NCk,n are unimodular triangulations of Ok,n ,

the previous lemma has the following corollary.

COROLLARY 4.20. Fix an a ∈ [k] and let t = (t1, . . . , tn−k) be a vector witha 6 t1 < t2 < · · · < tn−k 6 k(n − k) + a − k. From t we derive a partitionm = (m0, . . . ,mn−k) of k(n− k)− (n− k)− (n− k) by setting mb = tb+1− tb−1,with the conventions t0 = 0 and tn−k+1 = k(n − k)+ 1.

Then the following numbers coincide:

• The number of maximal faces of ∆NNk,n whose summing tableaux have their ath

row equal to t .

• The number of maximal faces of ∆NCk,n whose summing tableaux have their ath

row equal to t .

• The m-mixed volume of the tuple Oa,0k,n , . . . ,Oa,n−k

k,n .






Note that the inequalities a 6 t1 < t2 < · · · < tn−k 6 k(n − k) + a − k inthe corollary are necessary and sufficient for t to appear as the ath row in somestandard tableau.

EXAMPLE 4.21. Consider the case k = 2 and n = 5. The nonnesting andnoncrossing complexes each have five maximal faces, corresponding to thefollowing tableaux:

∆NN2,5 =

1 2 3

4 5 6

1 2 4

3 5 6

1 2 5

3 4 6

1 3 4

2 5 6

1 3 5

2 4 6

∆NC

2,5 = 1 2 3

2 4 6

1 2 4

2 5 6

1 2 5

3 5 6

1 3 4

3 4 6

1 3 5

4 5 6

As can be seen, the multisets of a-rows are the same in both complexes.

First rows = {123, 124, 125, 134, 135},Second rows = {246, 256, 356, 346, 456},

Of course, symmetry under exchange of k and n−k implies that the same happensfor columns:

First columns = {12, 12, 13, 13, 14},Second columns = {24, 25, 25, 34, 35},

Third columns = {36, 46, 46, 56, 56}.

5. Relation to the weak separation complex

5.1. The weak separation complex. Leclerc and Zelevinsky [LZ98]introduced the complex of weakly separated subsets of [n] and showed thatits faces are the sets of pairwise quasicommuting quantum Plucker coordinatesin a q-deformation of the coordinate ring of the flag variety respectively theGrassmannian. A geometric version of their definition is that two subsets X andY of [n] are weakly separated if, when considered as subsets of vertices in ann-gon, the convex hulls of X\Y and Y\X are disjoint. If we restrict to X and Y






of fixed size k then we are in the Grassmann situation and the following complexwas studied by Scott [Sco05, Sco06] as a subcomplex of the Leclerc–Zelevinskycomplex.

DEFINITION 5.1. Let I and J be two vectors in Vk,n . We say that I and J areweakly separated if, considered as subsets of vertices of an n-gon, the convexhulls of I\J and J\I do not meet. We denote by ∆Sep

k,n the simplicial complex ofsubsets of Vk,n whose elements are pairwise weakly separated.

Scott conjectured that ∆Sepk,n is pure of dimension k(n− k) and that it is strongly

connected (that is, its dual graph is connected). Both conjectures were shown tohold by Oh et al. [OPS15], for the first one see also Danilov et al. [DKK10,Proposition 5.9].

It is not hard to see that ∆Sepk,n is a subcomplex of ∆NC

k,n and it is trivial to observethat ∆Sep

k,n is invariant under cyclic (or, more strongly, dihedral) symmetry. Ournext results combines these two properties and states that ∆Sep

k,n is the ‘cyclic part’of ∆NC

k,n . In the following statement, we denote by I+i the cyclic shift of I ∈ Vk,n

by the amount i , that is, the image of I under the map x → x + i considered asremainders 1, . . . , n modulo n.

PROPOSITION 5.2. The weak separation complex∆Sepk,n is equal to the intersection

of all cyclic shifts of the noncrossing complex ∆NCk,n:

(1) If I and J are weakly separated, then they are noncrossing.

(2) If I+i and J+i are noncrossing for every i ∈ [n], then I and J are weaklyseparated.

Proof. Assume that 1 6 a < b 6 k are indices such that for (i1, . . . , ik) and ( j1,

. . . , jk) in Vk,n we have i` = j` for a < ` < b and (ia, ib) and ( ja, jb) cross. Thenwe must have ia, ib ∈ I\J and ja, jb ∈ J\I . Thus the convex hulls of I\J andJ\I intersect nontrivially and hence I and J cannot be weakly separated. Thiscompletes the proof of (1).

Next, recall that we have seen in Proposition 1.8(iii) that the crossing ornoncrossing of I and J only depends on I4J , as does weak separability bydefinition. Hence, there is no loss of generality in assuming that I and Jare complementary to one another. Think of I as a cyclic sequence of plusesand minuses, indicating horizontal and vertical steps when you look at I as amonotone path in the k×(n−k) grid (the complementarity assumption, of course,implies k = n − k), compare Figure 1 in the introduction. Then I and J areopposite sequences. Observe that I and its complement J are weakly separated






if and only if I changes signs (considered cyclically) exactly twice. In otherwords, I and J are weakly separated if and only if, after some cyclic shift, itconsists of n/2 pluses followed by n/2 minuses. Suppose that I is not of thatform, and it changes sign at least four times. Let a and b be the lengths of thefirst two maximal constant subsequences. We then are in one of the following twosituations.

• If a > b, shifting the sequence by a − b we get that I starts with two constantsubsequences of the same length b. This implies that I crosses its complement.

• If a < b, shifting the sequence by 2a we get that I finishes with two constantsubsequences of the same length a. This implies as well that I crosses itscomplement.

Statement (2) follows.

The fact that ∆Sepk,n is a subcomplex of ∆NC

k,n was already observed in [PPS10,Lemma 2.10]. Proposition 5.2 and the results from [OPS15] immediately implythe following corollary.

COROLLARY 5.3. ∆Sepk,n is a full-dimensional pure flag subcomplex of ∆NC

k,n .

5.2. A conjecture on the topology of ∆Sepk,n. The (cyclic) intervals considered

in Proposition 2.10(v) were shown to be vertices in all maximal faces of ∆NCk,n .

They are as well vertices in all maximal faces of ∆Sepk,n . Thus it makes sense

to look at the reduced weak separation complex ∆Sepk,n , which is a subcomplex

of ∆NCk,n . Observe that both complexes coincide for k = 2. This follows for

example from Proposition 5.2 and the fact that ∆NC2,n is cyclic symmetric. Hence,

∆Sep2,n is an (n − 4)-sphere, the dual associahedron. The topology of ∆Sep

k,n and itsgeneralizations motivated by the work of Leclerc and Zelevinsky [LZ98] hasbeen scrutinized in [HH11, HH13]. In the preliminary version [HH11] theymake detailed conjectures about the topology of the various complexes basedon computer experiments. In particular, they observe that for every k and n thecomplex ∆Sep

k,n appears to have the same homology as an (n − 4)-sphere. We statetheir conjecture in this case and remark that in the first version of the presentpaper, we had independently come to the same conjecture. We learnt about thepreliminary version [HH11] from Vic Reiner after the first version of the presentpaper was released on the arxiv.

CONJECTURE 5.4. For every 2 6 k 6 n− 2 the reduced complex ∆Sepk,n of weakly

separated k-subsets of [n] is homotopy equivalent to the (n − 4)-sphere.






Acknowledgements

The authors would like to thank Christian Haase and Raman Sanyal for valuablecomments and discussions. After the first version of this paper was submittedto the arXiv we became aware of the facts that our results substantially overlapwith [PPS10] and that Conjecture 5.4 was already posed in the preliminaryversion [HH11] of [HH13]. We thank David Speyer and Vic Reiner, respectively,for pointing these two facts to us. An extended abstract of this paper appeared inDMTCS as part of the FPSAC 2014 conference proceedings. F.S. was supportedby the Spanish Ministry of Science (MICINN) through grant MTM2011-22792,and by a Humboldt Research Award of the Alexander von Humboldt Foundation.C.S. was supported by the German Research Foundation DFG, grant STU563/2-1 ‘Coxeter-Catalan combinatorics’. We acknowledge support by theGerman Research Foundation and the OpenAccess Publication Funds of theFreie Universitat Berlin.

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noncrossing sets and a grassmann associahedron€¦ · noncrossing sets and a grassmann...

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